# Syllogism

Do not let the other decision maker fool you. The following article is the holy of holies in regard to deductive logic. It cannot be referenced to one individual because it is a plain abstract. However the specific model provided is traceable to a small Oxford school working around the late nineteenth century. Since only one Oxford school of logic exists at that precise time that is sufficient referencing.

As provided the model is entirely my own design and it is deliberately designed towards a metaphysical mystical articulation. That is, it is not meant to be understood as an instance. Instead it is meant to be wondered at as a pattern of universal nature.

Where the pattern as given is error checked for correctness it will be easy enough for any who can do so, to make any necessary corrections without changing the manner of its presentation. Before making any changes I would affirm that it seems correct as stated.

All of the nineteen Silly Gisms can be translated into the algebraic notation that I have presented in these articles. To do so, it is necessary to understand the following explanation as stated, not to be misled by the perversities of language use in terms of amphiboly, tautology, ambiguity, oxymorons, and other such enitity that tend to live their ghost like existence in this kind of domain.

Therefore when I use "All B is C" in this context I do not mean "All B equals C" in the sense of our understanding of equals and is, which we have looked at in other articles. Instead in this specific context, the word use must only be understood as follows: That is, if "All B is C" we mean draw a big circle and call it C. Inside the big circle called "C" draw a much smaller circle called "B". Now if you have done that you will easily notice that all B is C, but not all C is B. Since all B is C but not all C is B we cannot say "All B equals C", and in fact we can definitely state that the proposition is to be called false and not true. Because we could place another smaller circle inside "C" that is in no way connected to "B" and call it "A". Then you would have a large circle "C" and in the large circle are entirely contained two smaller circles called "A" and "B" where the two smaller circles are in no way connected.

Then we would have "All A is C" and "All B is C" and "No A is B" and "No B is A" and "Some C is A" and "Some C is B".

Therefore, in order to translate the nineteen Silly Gisms into the language of algebra given in these articles the genus/species relationship is used. Such that in our example given above, the large circle "C" in this instance is the <Genus> and the two smaller circles "A" and "B" are both <Species> of "C".

So the basic translation would be along the lines:

C < A and C < B => C < (A ^ B). If C contains A and C contains B then C contains both A and B.

However, because the patterns involve the use of the universal and existential operators in regard to the words "All" and "Some" therefore you would have to be confident about using the higher order logic signlabels "ψ" and "Э".

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.

The statement of the prime pattern in its basic form begins at this point, and is given as notes I through to VIII. The notes I - VIII are entirely the work of the late nineteenth century Oxford school of logic although the pattern as given and the manner of presentation is entirely my own.

I.

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.

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.

II.

Let C be the major term; B the middle term; A the minor term.

III.

A. All A is B.

E. No A is B.

I. Some A is B.

O. Some A is not B.

IV.

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.

V.

(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)

VI.

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.

VII.

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.

VIII.

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 statement of the prime pattern in its basic form is complete at this point, given as notes I through to VIII.

With the aim towards an intuitive grasp of the patterns and a commonplace familiarity with their shape and composition it is better to attend only to the first figure. Because the purpose of this article is to provide the above prime pattern given in notes I-VIII in its basic form. The possible diversities of explanation of the given pattern are so extensive that there is no requirement to attempt such in this context. Instead what I will do is describe a few lines of development that may be usefully followed in order to obtain the intuitive grasp and commonplace familiarity such that the patterns are embedded in themselves.

We do better to attend to the first figure exclusively because it is held to be the only real pattern of the four figures such that each of the other three figures refer back to the first figure. Also the first figure is most simple and straightforward so if we can clarify it properly it will inform us as to how better to understand the other three.

First Figure.

1. All B is C. All A is B. Therefore All A is C.

(ЭC < ψB) ^ (ЭB < ψA) => (ЭC < ψA)

If [Some C contains All B] and [Some B contains All A] is true, then: [Some C contains All A] is true.

2. No B is C. All A is B. Therefore No A is C.

(ψC < ~B) ^ (ЭB < ψA) => (ψC < ~A)

If [All C contains Not B] and [Some B contains All A] is true, then: [All C contains Not A] is true.

3. All B is C. Some A is B. Therefore Some A is C.

(ЭC < ψB) ^ (ЭB < ЭA) => (ЭC < ЭA)

If [Some C contains All B] and [Some B contains Some A] is true, then: [Some C contains Some A] is true.

4. No B is C. Some A is B. Therefore Some A is not C.

(ψC < ~B) ^ (ЭB < ЭA) => (~C < ЭA)

If [All C contains Not B] and [Some B contains Some A] is true, then: [Not C contains Some A] is true.

The First Figure is now detailed in three different signlabel language systems, where all three language systems are supposed to be saying the exact same thing. According to our method the requirement now is to challenge the correctness and truth of the stated patterns of the first figure. That is, can we satisfy ourselves that the three different language explanations are each saying what they are supposed to be saying? If they are, then do they each say the same thing as the other two in regard to the matter they refer towards? And is the matter they refer towards in each case correct, accurately stated and true?

First Pattern, First Figure.

For example the first pattern refers towards a matter that can be visually described as large circle "C" entirely contains smaller circle "B" and smaller circle "B" entirely contains little circle "A", which we believe justifies the claim that larger circle "C" entirely contains little circle "A". Which we then describe using three different signlabel language systems as follows:

1. All B is C. All A is B. Therefore All A is C.

(ЭC < ψB) ^ (ЭB < ψA) => (ЭC < ψA)

If [Some C contains All B] and [Some B contains All A] is true, then: [Some C contains All A] is true.

So the question we ask is whether the matter as described in terms of three circles is what is referred towards by the descriptions. And do the descriptions in each case say the same thing, in the correct way such that in the total understanding there is coherence and complete non-contradiction. As such we are asking a question about "meaning" and in particular do the meanings of the three language systems translate properly, so that we are confident that each proposition says something at all.

First Proposition, First Pattern, First Figure.

In order to clarify the "meaning" value of the pattern as a whole we must establish the equivalence of each individual proposition, as follows:

All B is C <=> [Some C contains All B] <=> (ЭC < ψB)

This is the first proposition of the first pattern of the first figure.

Our visual description is: {a large circle "C" which entirely contains a smaller circle "B"}.

To establish the meaning of this propostion it is useful to apply "Dodgson's Game" referred to in http://en.wikiversity.org/wiki/Game_domain_model as follows:

Draw a vertical line where the higher point of the line is designated "C" and the lower point of the line is designated "~C". And draw an horizontal line bisecting the vertical line where the right point of the line is designated "B" and the left point of the line is designated "~B". A white counter placed on the grid designates existant and a black counter placed on the grid designates non-existant.

Such that if we place a white counter directly on the vertical line at the upper half above the horizontal line we are saying this:

"Some C are B and Some C are Not B". Which is interesting because it positively states a necessary implication of our first proposition. The necessary implication of: "Some C contains B" is that "Some C contains Not B".

So we now have to go back to our picture of the two circles to see whether it says the same thing. And what we discover is that all of the area of the larger circle "C" that is not covered by the smaller circle "B" is called "Not B".

And to clarify what we are not saying, we are definitely not saying "All C contains B". If we were saying that, then in Dodgson's Game it would mean we placed a white counter at the upper right corner "CB". So we now have to look at the two possible placements of the white counter on the grid in order to determine that the meaning of the two propositions is quite different.

In this way we establish the particular meaning of our first proposition by demonstrating what it does not mean.

The next useful thing we discover through Dodgson's Game is that our placement of the white counter has only told us half of the first proposition. That is, it tells us that "Some C is B", but it does not tell us that "the only existant B are C". So with the use of only the white counter we do not know whether any "B Not C" exist. Whereas our first proposition definitely says that "C contains All B" meaning that "No B Not C" exist.

Therefore to show the complete first proposition we are required to place a black counter on the bottom right corner "B~C". This says the other half of the first proposition that "No B are Not C".

That means, through the Dodgson's Game we discover several interesting things about our first proposition, including that:

i. Some C are B implies that Some C are Not B.

ii. The claim that <All> B are C is two components, that is, "B are C" and "No B are Not C".

Therefore we arrive at an end point, that is several different things. Our first proposition is well stated as:

All B is C <=> [Some C contains All B] <=> (ЭC < ψB)

It has two necessary implications that could themselves be stated as independent propositions: One, that "Some C are Not B" and two, that "No B are Not C".

Second Proposition, First Pattern, First Figure.

All A is B <=> [Some B contains All A] <=> (ЭB < ψA)

This is the second proposition of the first pattern of the first figure.

Our visual description is: {a smaller circle "B" which entirely contains a little circle "A"}.

The argumentation that applied to the first proposition correlates to the argument that applies to the second proposition because the shape not meaning of the second proposition is similar to the first. We can prove this by comparing the two propositions within the same language as follows:

<All B is C> is similar in shape to <All A is B>.

<[Some C contains All B]> is similar in shape to <[Some B contains All A]>.

<(ЭC < ψB)> is similar in shape to <(ЭB < ψA)>.

Therefore, our second proposition is meaningful for two reasons.

One is that the structure of its meaning stands in as much as we have discovered the structure of the first proposition to be meaningfully correct and true, because the structure of both propositions is the same. Two is that the content of its meaning stands in as much as it is different content to the first proposition.

Third Proposition, First Pattern, First Figure.

All A is C <=> [Some C contains All A] <=> (ЭC < ψA)

This is the third proposition of the first pattern of the first figure. It is called the conclusion of the first pattern and is subject to measurement in terms of validity, truth and soundness.

Our visual description is: {a larger circle "C" which entirely contains a little circle "A"}.

We can first affirm that the shape of the third propositon is exactly similar to the first two, and that the content of each component is different to any other component:

<All B is C> :: <All A is B> :: <All A is C>

<[Some C contains All B]> :: <[Some B contains All A]> :: [Some C contains All A]

<(ЭC < ψB)> :: <(ЭB < ψA)> :: <(ЭC < ψA)>

In regards to measurement, by validity we ask whether the conclusion as stated is a necessary consequence of the first two propositions. In order to qualify as a valid conclusion it does not matter whether the first two propositions are imaginary or real, fictitious or fact, lies or truth. A valid conclusion is only one that meets the condition of necessity. That is, where the conclusion is a necessary consequence of the propositions it is called valid, regardless of whether it is true or false.

A conclusion that is called sound must meet two specific conditions. The first is that the argument must be valid in accord with the description provided. The second is that the two original propositions called the major and minor premisses must both be individually true. If the two premisses are both true and the argument from the premisses is valid then the conclusion is called sound. Furthermore, a valid argument from true premisses guarantees that the conclusion is itself true. That is, it is not possible to generate a false conclusion from a valid argument with true premisses.

For this reason it is easy enough to understand that the conclusion to the first pattern cannot be called sound in the abstract form. That is because the pattern itself must equally support argument that is imaginary, fictitious, lies and falsity as much as it supports argument that is real, fact, honest and true.

The only requirement is that the conclusion be a necessary consequence of the premisses.

Such being the case, we must attend to the simple question:

Is it a necessary consequence that C contains A, if our first two premisses are that B contains A and C contains B?

The nature of this problem is disguised and it is not of the same heirarchical order as the establishment of the meaning of the propositions. If we take it that we are satisfied with our establishment of meaning of all three propositions, as stated. Which I would do since the explanation as given is sufficient for the purposes of the article.

Then reasonably we must notice that this does nothing at all to establish the necessary consequence of the conclusion from the premisses. The immediately previous statement is not to claim that the conclusion is not a necessary consequence of the premisses. Only that we have as yet no proof that it is so.

Unfortunately that side of the matter is very odd and not the focus of attention in this particular discourse. In order to explain some of its nature to satisfy our imagination in the first instance this idea:

We have a box called C. Inside the box C we place a smaller box B. Inside box B we place a smaller box A. Therefore we make the claim box C contains box A. But no it does not. Show me a box C with a box A without a box B. Well no, we cannot do that, because box A is inside box B. Then inside box B and not inside box C. Well no, inside box C because box B is inside box C. What if box B was not inside box C? Well, box A would still be inside box B. Proof then that box A is inside box B regardless of whether box B is inside box C or not. Okay, so what if I take box B out of box C and box A out of box B. And then I place box A inside box C? Yes that is fine. Because now box C contains box A. The only problem with it is you have lost both our Major and Minor Premisses. Because our major premiss was box C contains box B and now it doesn't. And our minor premiss was box B contains box A and now it doesn't.

And this is my explanation. That the conclusion C contains A is the description of box C containing box A without box C containing box B or box B containing box A. Which is not what we mean. For this reason I would notice a particular puzzle which we can label as the "Problem of Necessary Consequence". All we need do at this stage is acknowledge that the problem of necessary consequence exists, recognise that we are not required to solve it at this stage, and make a mental note to label it whenever we find it in any of the later patterns. And for the sake of argument refer it to the great god David Hume for his immediate attention.

Second Pattern, First Figure.

The second pattern refers to a matter that can be visually described as a large square box called "U" which means Universe of Discourse. Inside "U" are two entirely separate circles, where neither circle is cut by the circumference of the other circle. One circle is called by the namelabel "C" and the other circle by the namelabel "B". Inside circle "B" is entirely contained a smaller circle "A". This we believe describes a circumstance where C is not B, and where B contains All A, which justifies us in believing that C is not A. Which we then describe using three different signlabel language systems as follows:

No B is C. All A is B. Therefore No A is C.

(ψC < ~B) ^ (ЭB < ψA) => (ψC < ~A)

If [All C contains Not B] and [Some B contains All A] is true, then: [All C contains Not A] is true.

So the question we ask is whether the matter as described in terms of three circles is what is referred towards by the descriptions. And do the descriptions in each case say the same thing, in the correct way such that in the total understanding there is coherence and complete non-contradiction. As such we are asking a question about "meaning" and in particular do the meanings of the three language systems translate properly, so that we are confident that each proposition says something at all.

To this end, and given the requirement that any individual so inclined can do the workings out for themselves given the previous explanation, all I would do here is clarify the equivalence of meaning between each of the languages for the three propositions separately, as follows:

Major Premiss: No B is C <=> [All C contains Not B] <=> (ψC < ~B)

Minor Premiss: All A is B <=> [Some B contains All A] <=> (ЭB < ψA)

Conclusion: No A is C <=> [All C contains Not A] <=> (ψC < ~A)

Third Pattern, First Figure.

The third pattern refers to a matter that can be visually described as a square shape "C" where the upper right quarter of the square shape is shaded in and called "B". That demonstrates that the square C entirely contains the square B as one of its corners. A third square the same size as "C" and called by the label "A" has a small part of one of its corners overlap half of square B. This we believe describes a circumstance where C entirely contains B and B contains some A, such that we are justified in believing that C contains some A. Which we then describe using three different signlabel language systems as follows:

All B is C. Some A is B. Therefore Some A is C.

(ЭC < ψB) ^ (ЭB < ЭA) => (ЭC < ЭA)

If [Some C contains All B] and [Some B contains Some A] is true, then: [Some C contains Some A] is true.

And then demonstrate in regard to equivalence of meaning between the different languages as:

Major Premiss: All B is C <=> [Some C contains All B] <=> (ЭC < ψB)

Minor Premiss: Some A is B <=> [Some B contains Some A] <=> (ЭB < ЭA)

Conclusion: Some A is C <=> [Some C contains Some A] <=> (ЭC < ЭA)

Fourth Pattern, First Figure.

The fourth pattern refers to a matter that can be visually described as two circles that are entirely separate such that the circumference of each in no way cuts the circumference of the other. One circle we call "B" and the other circle we call "C". This we believe describes the proposition that "No B is C". A third circle is drawn such that it partially overlaps the circle "B" so that the circumference of circle B cuts the circumference of the third circle which we call "A". This describes the proposition that Some A is B. And since No B is C, and some of A is B, we believe this justifies the proposition that some A is not C. Which we then describe using three different signlabel language systems as follows:

4. No B is C. Some A is B. Therefore Some A is not C.

(ψC < ~B) ^ (ЭB < ЭA) => (~C < ЭA)

If [All C contains Not B] and [Some B contains Some A] is true, then: [Not C contains Some A] is true.

And then demonstrate in regard to equivalence of meaning between the different languages as:

Major Premiss: No B is C <=> [All C contains Not B] <=> (ψC < ~B)

Minor Premiss: Some A is B <=> [Some B contains Some A] <=> (ЭB < ЭA)

Conclusion: Some A is not C <=> [Not C contains Some A] <=> (~C < ЭA)

Our position now is that we can describe the First Figure in four different medium or languages, where each medium apparently says the exact same thing. Which we should now show separately, as follows:

First Figure In Four Different Medium.

First Figure, Visually.

1. The first pattern refers towards a matter that can be visually described as large circle "C" entirely contains smaller circle "B" and smaller circle "B" entirely contains little circle "A", which we believe justifies the claim that larger circle "C" entirely contains little circle "A".

2. The second pattern refers to a matter that can be visually described as a large square box called "U" which means Universe of Discourse. Inside "U" are two entirely separate circles, where neither circle is cut by the circumference of the other circle. One circle is called by the namelabel "C" and the other circle by the namelabel "B". Inside circle "B" is entirely contained a smaller circle "A". This we believe describes a circumstance where C is not B, and where B contains All A, which justifies us in believing that C is not A.

3. The third pattern refers to a matter that can be visually described as a square shape "C" where the upper right quarter of the square shape is shaded in and called "B". That demonstrates that the square C entirely contains the square B as one of its corners. A third square the same size as "C" and called by the label "A" has a small part of one of its corners overlap half of square B. This we believe describes a circumstance where C entirely contains B and B contains some A, such that we are justified in believing that C contains some A.

4. The fourth pattern refers to a matter that can be visually described as two circles that are entirely separate such that the circumference of each in no way cuts the circumference of the other. One circle we call "B" and the other circle we call "C". This we believe describes the proposition that "No B is C". A third circle is drawn such that it partially overlaps the circle "B" so that the circumference of circle B cuts the circumference of the third circle which we call "A". This describes the proposition that Some A is B. And since No B is C, and some of A is B, we believe this justifies the proposition that some A is not C.

First Figure, Verbally.

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.

First Figure, Written Word.

1. If [Some C contains All B] and [Some B contains All A] is true, then: [Some C contains All A] is true.

2. If [All C contains Not B] and [Some B contains All A] is true, then: [All C contains Not A] is true.

3. If [Some C contains All B] and [Some B contains Some A] is true, then: [Some C contains Some A] is true.

4. If [All C contains Not B] and [Some B contains Some A] is true, then: [Not C contains Some A] is true.

First Figure, Algebraically.

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)

At this stage we have adjusted our understanding of the First Figure through a range of different explanations in various languages. Our better move now is to take a reversed sequence of explanation, because we understand what we want to obtain in this instance. By which I mean we should specifically detail the nineteen patterns algebraically first. Having the nineteen patterns algebraically stated, we should then provide the written word form for the same patterns. And having the written word form we should again detail the verbal word patterns, which should be exactly the same statement as they are presently stated. In this manner we are starting as if the algebraic patterns are considered correct, we are then translating them into the written word form to see if this confirms our understanding and describing the verbal explanation to determine if it is the same as it presently is. Therefore our next task is to provide the nineteen algebraic patterns, which must follow the exact same format as the first four as already given.

The Nineteen Algebraic Patterns.

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)