Euler Diagram

This article is about Eulerian circles of set theory and logic. For the geometric Euler circle, see Nine-point circle.

An Euler diagram illustrating that the set of "animals with four legs" is a subset of "animals", but the set of "minerals" is disjoint (has no members in common) with "animals".

An Euler diagram is a diagrammatic means of representing sets and their relationships. The first use of "Eulerian circles" is commonly attributed to Swiss mathematician Leonhard Euler (1707–1783). They are closely related to Venn diagrams.

Venn and Euler diagrams were incorporated as part of instruction in set theory as part of the new math movement in the 1960s. Since then, they have also been adopted by other curriculum fields such as reading.^[1]

Contents 1 Overview 2 History 3 Example: Euler- to Venn-diagram and Karnaugh map 4 Gallery 5 Footnotes 6 References 7 External links

Overview

Euler diagrams consist of simple closed curves (usually circles) in the plane that depict sets. The sizes or shapes of the curves are not important: the significance of the diagram is in how they overlap. The spatial relationships between the regions bounded by each curve (overlap, containment or neither) corresponds to set-theoretic relationships (intersection, subset and disjointness).
Each Euler curve divides the plane into two regions or "zones": the interior, which symbolically represents the elements of the set, and the exterior, which represents all elements that are not members of the set. Curves whose interior zones do not intersect represent disjoint sets. Two curves whose interior zones intersect represent sets that have common elements; the zone inside both curves represents the set of elements common to both sets (the intersection of the sets). A curve that is contained completely within the interior zone of another represents a subset of it.

Examples of small Venn diagrams (on left) with shaded regions representing empty sets, showing how they can be easily transformed into equivalent Euler diagrams (right).

Venn diagrams are a more restrictive form of Euler diagrams. A Venn diagram must contain all the possible zones of overlap between its curves, representing all combinations of inclusion/exclusion of its constituent sets, but in an Euler diagram some zones might be missing. When the number of sets grows beyond 3, or even with three sets, but under the allowance of more than two curves passing at the same point, we start seeing the appearance of multiple mathematically unique Venn diagrams. Venn diagrams represent the relationships between n sets, with 2ⁿ zones, Euler diagrams may not have all zones. (An example is given below in the History section; in the top-right illustration the O and I diagrams are merely rotated; Venn stated that this difficulty in part led him to develop his diagrams).

In a logical setting, one can use model theoretic semantics to interpret Euler diagrams, within a universe of discourse. In the examples above, the Euler diagram depicts that the sets Animal and Mineral are disjoint since the corresponding curves are disjoint, and also that the set Four Legs is a subset of the set of Animals. The Venn diagram, which uses the same categories of Animal, Mineral, and Four Legs, does not encapsulate these relationships. Traditionally the emptiness of a set in Venn diagrams is depicted by shading in the region. Euler diagrams represent emptiness either by shading or by the use of a missing region.
Often a set of well-formedness conditions are imposed; these are topological or geometric constraints imposed on the structure of the diagram. For example, connectedness of zones might be enforced, or concurrency of curves or multiple points might be banned, as might tangential intersection of curves. In the diagram to the right, examples of small Venn diagrams are transformed into Euler diagrams by sequences of transformations; some of the intermediate diagrams have concurrency of curves. However, this sort of transformation of a Venn diagram with shading into an Euler diagram without shading is not always possible. There are examples of Euler diagrams with 9 sets that are not drawable using simple closed curves without the creation of unwanted zones since they would have to have non-planar dual graphs.
 

History

 

Photo of page from Hamilton's 1860 "Lectures" page 180. (Click on it, up to two times, to enlarge). The symbolism A, E, I, and O refer to the four forms of the syllogism. The small text to the left says: "The first employment of circular diagrams in logic improperly ascribed to Euler. To be found in Christian Weise."

On the right is a photo of page 74 from Couturat 1914 wherein he labels the 8 regions of the Venn diagram. The modern name for these "regions" is minterms. These are shown on the left with the variables x, y and z per Venn's drawing. The symbolism is as follows: logical AND ( & ) is represented by arithmetic multiplication, and the logical NOT ( ~ )is represented by " ' " after the variable, e.g. the region x'y'z is read as "NOT x AND NOT y AND z" i.e. ~x & ~y & z.

Both the Veitch and Karnaugh diagrams show all the minterms, but the Veitch is not particularly useful for reduction of formulas. Observe the strong resemblance between the Venn and Karnaugh diagrams; the colors and the variables x, y, and z are per Venn's example.

As shown in the illustration to the right, Sir William Hamilton in his posthumously published Lectures on Metaphysics and Logic (1858–60) asserts that the original use of circles to "sensualize ... the abstractions of Logic" (p. 180) was not Leonhard Paul Euler (1707–1783) but rather Christian Weise (?–1708) in his Nucleus Logicoe Weisianoe that appeared in 1712 posthumously. He references Euler's Letters to a German Princess on different Matters of Physics and Philosophy¹" [¹Partie ii., Lettre XXXV., ed. Cournot. – ED.]^[2]
In Hamilton's illustration the four forms of the syllogism as symbolized by the drawings A, E, I and O are:^[3]

• A: The Universal Affirmative, Example: "All metals are elements".
• E: The Universal Negative, Example: "No metals are compound substances".
• I: The Particular Affirmative, Example: "Some metals are brittle".
• O: The Particular Negative, Example: "Some metals are not brittle".

In his 1881 Symbolic Logic Chapter V "Diagrammatic Representation", John Venn (1834–1923) comments on the remarkable prevalence of the Euler diagram:

"...of the first sixty logical treatises, published during the last century or so, which were consulted for this purpose:-somewhat at random, as they happened to be most accessible :-it appeared that thirty four appealed to the aid of diagrams, nearly all of these making use of the Eulerian Scheme." (Footnote 1 page 100)

Composite of two pages 115–116 from Venn 1881 showing his example of how to convert a syllogism of three parts into his type of diagram. Venn calls the circles "Eulerian circles" (cf Sandifer 2003, Venn 1881:114 etc) in the "Eulerian scheme" (Venn 1881:100) of "old-fashioned Eulerian diagrams" (Venn 1881:113).

But nevertheless, he contended "the inapplicability of this scheme for the purposes of a really general Logic" (page 100) and in a footnote observed that "it fits in but badly even with the four propositions of the common Logic [the four forms of the syllogism] to which it is normally applied" (page 101). Venn ends his chapter with the observation that will be made in the examples below – that their use is based on practice and intuition, not on a strict algorithmic practice:

“In fact ... those diagrams not only do not fit in with the ordinary scheme of propositions which they are employed to illustrate, but do not seem to have any recognized scheme of propositions to which they could be consistently affiliated.” (pp. 124–125)

Finally, in his Chapter XX HISTORIC NOTES Venn gets to a crucial criticism (italicized in the quote below); observe in Hamilton's illustration that the O (Particular Negative) and I (Particular Affirmative) are simply rotated:

"We now come to Euler's well-known circles which were first described in his Lettres a une Princesse d'Allemagne (Letters 102–105). The weak point about these consists in the fact that they only illustrate in strictness the actual relations of classes to one another, rather than the imperfect knowledge of these relations which we may possess, or wish to convey, by means of the proposition. Accordingly they will not fit in with the propositions of common logic, but demand the constitution of a new group of appropriate elementary propositions.... This defect must have been noticed from the first in the case of the particular affirmative and negative, for the same diagram is commonly employed to stand for them both, which it does indifferently well". (italics added: page 424)

(Sandifer 2003 reports that Euler makes such observations too; Euler reports that his figure 45 (a simple intersection of two circles) has 4 different interpretations). Whatever the case, armed with these observations and criticisms, Venn then demonstrates (pp. 100–125) how he derived what has become known as his Venn diagrams from the "old-fashioned Euler diagrams". In particular he gives an example, shown on the left.
By 1914 Louis Couturat (1868–1914) had labeled the terms as shown on the drawing on the right. Moreover, he had labeled the exterior region (shown as a'b'c') as well. He succinctly explains how to use the diagram – one must strike out the regions that are to vanish:

"VENN'S method is translated in geometrical diagrams which represent all the constituents, so that, in order to obtain the result, we need only strike out (by shading) those which are made to vanish by the data of the problem." (italics added p. 73)

Given the Venn's assignments, then, the unshaded areas inside the circles can be summed to yield the following equation for Venn's example:

"No Y is Z and ALL X is Y: therefore No X is Z" has the equation x'yz' + xyz' + x'y'z for the unshaded area inside the circles (but note that this is not entirely correct; see the next paragraph).

In Venn the 0th term, x'y'z', i.e. the background surrounding the circles, does not appear. Nowhere is it discussed or labeled, but Couturat corrects this in his drawing. The correct equation must include this unshaded area shown in boldface:

"No Y is Z and ALL X is Y: therefore No X is Z" has the equation x'yz' + xyz' + x'y'z + x'y'z' .

In modern usage the Venn diagram includes a "box" that surrounds all the circles; this is called the universe of discourse or the domain of discourse.
Couturat now observes that, in a direct algorithmic (formal, systematic) manner, one cannot derive reduced Boolean equations, nor does it show how to arrive at the conclusion "No X is Z". Couturat concluded that the process "has ... serious inconveniences as a method for solving logical problems":

"It does not show how the data are exhibited by canceling certain constituents, nor does it show how to combine the remaining constituents so as to obtain the consequences sought. In short, it serves only to exhibit one single step in the argument, namely the equation of the problem; it dispenses neither with the previous steps, i. e., "throwing of the problem into an equation" and the transformation of the premises, nor with the subsequent steps, i. e., the combinations that lead to the various consequences. Hence it is of very little use, inasmuch as the constituents can be represented by algebraic symbols quite as well as by plane regions, and are much easier to deal with in this form."(p. 75)

Thus the matter would rest until 1952 when Maurice Karnaugh (1924– ) would adapt and expand a method proposed by Edward W. Veitch; this work would rely on the truth table method precisely defined in Emil Post's 1921 PhD thesis "Introduction to a general theory of elementary propositions" and the application of propositional logic to switching logic by (among others) Claude Shannon, George Stibitz, and Alan Turing.^[4] For example, in chapter "Boolean Algebra" Hill and Peterson (1968, 1964) present sections 4.5ff "Set Theory as an Example of Boolean Algebra" and in it they present the Venn diagram with shading and all. They give examples of Venn diagrams to solve example switching-circuit problems, but end up with this statement:

"For more than three variables, the basic illustrative form of the Venn diagram is inadequate. Extensions are possible, however, the most convenient of which is the Karnaugh map, to be discussed in Chapter 6." (p. 64)

In Chapter 6, section 6.4 "Karnaugh Map Representation of Boolean Functions" they begin with:

"The Karnaugh map¹ [¹Karnaugh 1953] is one of the most powerful tools in the repertory of the logic designer. ... A Karnaugh map may be regarded either as a pictorial form of a truth table or as an extension of the Venn diagram." (pp. 103–104)

The history of Karnaugh's development of his "chart" or "map" method is obscure. Karnaugh in his 1953 referenced Veitch 1951, Veitch referenced Claude E. Shannon 1938 (essentially Shannon's Master's thesis at M.I.T.), and Shannon in turn referenced, among other authors of logic texts, Couturat 1914. In Veitch's method the variables are arranged in a rectangle or square; as described in Karnaugh map, Karnaugh in his method changed the order of the variables to correspond to what has become known as (the vertices of) a hypercube.

Example: Euler- to Venn-diagram and Karnaugh map

This example shows the Euler and Venn diagrams and Karnaugh map deriving and verifying the deduction "No X's are Z's". In the illustration and table the following logical symbols are used:

1 can be read as "true", 0 as "false"

~ for NOT and abbreviated to ' when illustrating the minterms e.g. x' =_defined NOT x,

+ for Boolean OR (from Boolean algebra: 0+0=0, 0+1 = 1+0 = 1, 1+1=1)

& (logical AND) between propositions; in the mintems AND is omitted in a manner similar to arithmetic multiplication: e.g. x'y'z =_defined ~x & ~y & z (From Boolean algebra: 0*0=0, 0*1 = 1*0=0, 1*1 = 1, where * is shown for clarity)

→ (logical IMPLICATION): read as IF ... THEN ..., or " IMPLIES ", P → Q =_defined NOT P OR Q

Before it can be presented in a Venn diagram or Karnaugh Map, the Euler diagram's syllogism "No Y is Z, All X is Y" must first be reworded into the more formal language of the propositional calculus: " 'It is not the case that: Y AND Z' AND 'If an X then a Y' ". Once the propositions are reduced to symbols and a propositional formula ( ~(y & z) & (x → y) ), one can construct the formula's truth table; from this table the Venn and/or the Karnaugh map are readily produced. By use of the adjacency of "1"s in the Karnaugh map (indicated by the grey ovals around terms 0 and 1 and around terms 2 and 6) one can "reduce" the example's Boolean equation i.e. (x'y'z' + x'y'z) + (x'yz' + xyz') to just two terms: x'y' + yz'. But the means for deducing the notion that "No X is Z", and just how the reduction relates to this deduction, is not forthcoming from this example.

Given a proposed conclusion such as "No X is a Z", one can test whether or not it is a correct deduction by use of a truth table. The easiest method is put the starting formula on the left (abbreviate it as "P") and put the (possible) deduction on the right (abbreviate it as "Q") and connect the two with logical implication i.e. P → Q, read as IF P THEN Q. If the evaluation of the truth table produces all 1's under the implication-sign (→, the so-called major connective) then P → Q is a tautology. Given this fact, one can "detach" the formula on the right (abbreviated as "Q") in the manner described below the truth table.
Given the example above, the formula for the Euler and Venn diagrams is:

"No Y's are Z's" and "All X's are Y's": ( ~(y & z) & (x → y) ) =_defined P

And the proposed deduction is:

"No X's are Z's": ( ~ (x & z) ) =_defined Q

So now the formula to be evaluated can be abbreviated to:

( ~(y & z) & (x → y) ) → ( ~ (x & z) ): P → Q

IF ( "No Y's are Z's" and "All X's are Y's" ) THEN ( "No X's are Z's" )

The Truth Table demonstrates that the formula ( ~(y & z) & (x → y) ) → ( ~ (x & z) ) is a tautology as shown by all 1's in yellow column..

Square #	Venn, Karnaugh region		x	y	z		(~	(y	&	z)	&	(x	→	y))	→	(~	(x	&	z))
0	x'y'z'		0	0	0		1	0	0	0	1	0	1	0	1	1	0	0	0
1	x'y'z		0	0	1		1	0	0	1	1	0	1	0	1	1	0	0	1
2	x'yz'		0	1	0		1	1	0	0	1	0	1	1	1	1	0	0	0
3	x'yz		0	1	1		0	1	1	1	0	0	1	1	1	1	0	0	1
4	xy'z'		1	0	0		1	0	0	0	0	1	0	0	1	1	1	0	0
5	xy'z		1	0	1		1	0	0	1	0	1	0	0	1	0	1	1	1
6	xyz'		1	1	0		1	1	0	0	1	1	1	1	1	1	1	0	0
7	xyz		1	1	1		0	1	1	1	0	1	1	1	1	0	1	1	1

At this point the above implication P → Q (i.e. ~(y & z) & (x → y) ) → ~(x & z) ) is still a formula, and the deduction – the "detachment" of Q out of P → Q – has not occurred. But given the demonstration that P → Q is tautology, the stage is now set for the use of the procedure of modus ponens to "detach" Q: "No X's are Z's" and dispense with the terms on the left.^[5]
Modus ponens (or "the fundamental rule of inference"^[6]) is often written as follows: The two terms on the left, "P → Q" and "P", are called premises (by convention linked by a comma), the symbol ⊢ means "yields" (in the sense of logical deduction), and the term on the right is called the conclusion:

P → Q, P ⊢ Q

For the modus ponens to succeed, both premises P → Q and P must be true. Because, as demonstrated above the premise P → Q is a tautology, "truth" is always the case no matter how x, y and z are valued, but "truth" will only be the case for P in those circumstances when P evaluates as "true" (e.g. rows 0 OR 1 OR 2 OR 6: x'y'z' + x'y'z + x'yz' + xyz' = x'y' + yz').^[7]

P → Q , P ⊢ Q

i.e.: ( ~(y & z) & (x → y) ) → ( ~ (x & z) ) , ( ~(y & z) & (x → y) ) ⊢ ( ~ (x & z) )

i.e.: IF "No Y's are Z's" and "All X's are Y's" THEN "No X's are Z's", "No Y's are Z's" and "All X's are Y's" ⊢ "No X's are Z's"

One is now free to "detach" the conclusion "No X's are Z's", perhaps to use it in a subsequent deduction (or as a topic of conversation).
The use of tautological implication means that other possible deductions exist besides "No X's are Z's"; the criterion for a successful deduction is that the 1's under the sub-major connective on the right include all the 1's under the sub-major connective on the left (the major connective being the implication that results in the tautology). For example, in the truth table, on the right side of the implication (→, the major connective symbol) the bold-face column under the sub-major connective symbol " ~ " has the all the same 1s that appear in the bold-faced column under the left-side sub-major connective & (rows 0, 1, 2 and 6), plus two more (rows 3 and 4).

Gallery

• 

  A Venn diagram shows all possible intersections.
• 

  Euler diagram visualizing a real situation, the relationships between various supranational European organisations.
• 

  Euler diagram visualizing a real situation, the relationships between various supranational African organisations.
• 

  Humorous diagram comparing Euler and Venn diagrams.
• 

  Euler diagram of types of triangles, assuming isosceles triangles have at least 2 equal sides.
• 

  Euler diagram of terminology of the British Isles.

Footnotes

1. ^ Strategies for Reading Comprehension Venn Diagrams
2. ^ By the time these lectures of Hamilton were published, Hamilton too had died. His editors (symbolized by ED.), responsible for most of the footnoting, were the logicians Henry Longueville Mansel and John Veitch.
3. ^ Hamilton 1860:179. The examples are from Jevons 1881:71ff.
4. ^ See footnote at George Stibitz.
5. ^ This is a sophisticated concept. Russell and Whitehead (2nd edition 1927) in their Principia Mathematica describe it this way: "The trust in inference is the belief that if the two former assertions [the premises P, P→Q ] are not in error, the final assertion is not in error . . . An inference is the dropping of a true premiss [sic]; it is the dissolution of an implication" (p. 9). Further discussion of this appears in "Primitive Ideas and Propositions" as the first of their "primitive propositions" (axioms): *1.1 Anything implied by a true elementary proposition is true" (p. 94). In a footnote the authors refer the reader back to Russell's 1903 Principles of Mathematics §38.
6. ^ cf Reichenbach 1947:64
7. ^ Reichenbach discusses the fact that the implication P → Q need not be a tautology (a so-called "tautological implication"). Even "simple" implication (connective or adjunctive) will work, but only for those rows of the truth table that evaluate as true, cf Reichenbach 1947:64–66.

References

By date of publishing:

• Sir William Hamilton 1860 Lectures on Metaphysics and Logic edited by Henry Longueville Mansel and John Veitch, William Blackwood and Sons, Edinburgh and London.
• W. Stanley Jevons 1880 Elemetnary Lessons in Logic: Deductive and Inductive. With Copious Questions and Examples, and a Vocabulary of Logical Terms, M. A. MacMillan and Co., London and New York.
• John Venn 1881 Symbolic Logic, MacMillan and Co., London.
• Alfred North Whitehead and Bertrand Russell 1913 1st edition, 1927 2nd edition Principia Mathematica to *56 Cambridge At The University Press (1962 edition), UK, no ISBN.
• Louis Couturat 1914 The Algebra of Logic: Authorized English Translation by Lydia Gillingham Robinson with a Preface by Philip E. B. Jourdain, The Open Court Publishing Company, Chicago and London.
• Emil Post 1921 "Introduction to a general theory of elementary propositions" reprinted with commentary by Jean van Heijenoort in Jean van Heijenoort, editor 1967 From Frege to Gödel: A Sourcebook of Mathematical Logic, 1879–1931, Harvard University Press, Cambridge, MA, ISBN 0-674-42449-8 (pbk.)
• Claude E. Shannon 1938 "A Symbolic Analysis of Relay and Switching Circuits", Transactions American Institute of Electrical Engineers vol 57, pp. 471–495. Derived from Claude Elwood Shannon: Collected Papers edited by N.J.A. Solane and Aaron D. Wyner, IEEE Press, New York.
• Hans Reichenbach 1947 Elements of Symbolic Logic republished 1980 by Dover Publications, Inc., NY, ISBN 0-486-24004-5.
• Edward W. Veitch 1952 "A Chart Method for Simplifying Truth Functions", Transactions of the 1952 ACM Annual Meeting, ACM Annual Conference/Annual Meeting "Pittsburgh", ACM, NY, pp. 127–133.
• Maurice Karnaugh November 1953 The Map Method for Synthesis of Combinational Logic Circuits, AIEE Committee on Technical Operations for presentation at the AIEE summer General Meeting, Atlantic City, N. J., June 15–19, 1953, pp. 593–599.
• Frederich J. Hill and Gerald R. Peterson 1968, 1974 Introduction to Switching Theory and Logical Design, John Wiley & Sons NY, ISBN 0-71-39882-9.
• Ed Sandifer 2003 How Euler Did It, http://www.maa.org/editorial/euler/How%20Euler%20Did%20It%2003%20Venn%20Diagrams.pdf

External links

Wikimedia Commons has media related to: Euler diagrams

• Euler Diagrams. Brighton, UK (2004).What are Euler Diagrams?
• Visualisation with Euler diagrams project

Geometry Leads us to the Truth?

"The end he (the artist) strives for is something else than a perfectly executed print. His aim is to depict dreams, ideas, or problems in such a way that other people can observe and consider them." - M.C. Escher

I too have always been interested at the idea of what we can see deeper then what we observe on the surface. As if an abstraction in the geometry may be leading when considering Polytopes and allotrope s or even Penrose Tilings as to the Truth?:)

A remarkable mosaic of atoms

In quasicrystals, we find the fascinating mosaics of the Arabic world reproduced at the level of atoms: regular patterns that never repeat themselves. However, the configuration found in quasicrystals was considered impossible, and Dan Shechtman had to fight a fierce battle against established science. The Nobel Prize in Chemistry 2011 has fundamentally altered how chemists conceive of solid matter. See: The Nobel Prize in Chemistry 2011 Dan Shechtman

I do not think one can ever imagine what goes through my mind and I guess that's part of my artistic journey is to better learn how to describe what I am seeing. It goes back some time as to what I learn about myself and how some of these geometers see. I did not ever feel apart from them as I tried to look deeper into reality and see what the basis is and how  we might describe that.

You must also know I now sport an interesting tattoo that I will share shortly. Maybe even consider it as a line break, and as a pointer. You'll see why when I upload picture. So,  that has been my thing when I look at all this science and those espouse the teaching of,  that I tried to find my place in it. I mean I could be so wrong in a long of things.....but isn't that part of the evolution of being?  Learning about those mistakes and dealing with the responsibility of finding that truth within self?

If the heart was free from the impurities of sin, and therefore lighter than the feather, then the dead person could enter the eternal afterlife.

My second tattoo will be as in the picture showing below on this blog site demonstrating and seen above is an ancient idea about "our heart" in relation to "the truth."  How we weight that against one another and how the choices we make will have us asking whether we acted in accordance with that truth. That is "the final judgement" and if this is understood then we can access whether or not we have much more to learn. I know that setting right past mistakes is not an easy thing but if you at least start then that is part of the success of not of having to repeat them. Maybe repeat many times until you finally actually get it.

Well then,how does one simplify that picture of such Judgement in the Hall of Ma'at as to know that this message is alive and well in today's world and just as valid? How well will the tattooist portray this design? I'll have to give it to her  so she has some time to look at it and decipher.:)

 See Also:

• Fulleranes and Allotropes
• Coxeter and Plato's Cave
  

Trivium:Three Roads

 

Logic is the art of thinking; grammar, the art of inventing symbols and combining them to express thought; and rhetoric, the art of communicating thought from one mind to another, the adaptation of language to circumstance.Sister Miriam Joseph

 Painting by Cesare Maccari (1840-1919), Cicero Denounces Catiline.

In medieval universities, the trivium comprised the three subjects taught first: grammar, logic, and rhetoric. The word is a Latin term meaning “the three ways” or “the three roads” forming the foundation of a medieval liberal arts education. This study was preparatory for the quadrivium. The trivium is implicit in the De nuptiis of Martianus Capella, although the term was not used until the Carolingian era when it was coined in imitation of the earlier quadrivium.^[1] It was later systematized in part by Petrus Ramus as an essential part of Ramism.

Formal grammar

A formal grammar (sometimes simply called a grammar) is a set of rules of a specific kind, for forming strings in a formal language. The rules describe how to form strings from the language's alphabet that are valid according to the language's syntax. A grammar does not describe the meaning of the strings or what can be done with them in whatever context —only their form.

Formal language theory, the discipline which studies formal grammars and languages, is a branch of applied mathematics. Its applications are found in theoretical computer science, theoretical linguistics, formal semantics, mathematical logic, and other areas.

A formal grammar is a set of rules for rewriting strings, along with a "start symbol" from which rewriting must start. Therefore, a grammar is usually thought of as a language generator. However, it can also sometimes be used as the basis for a "recognizer"—a function in computing that determines whether a given string belongs to the language or is grammatically incorrect. To describe such recognizers, formal language theory uses separate formalisms, known as automata theory. One of the interesting results of automata theory is that it is not possible to design a recognizer for certain formal languages.

Parsing is the process of recognizing an utterance (a string in natural languages) by breaking it down to a set of symbols and analyzing each one against the grammar of the language. Most languages have the meanings of their utterances structured according to their syntax—a practice known as compositional semantics. As a result, the first step to describing the meaning of an utterance in language is to break it down part by part and look at its analyzed form (known as its parse tree in computer science, and as its deep structure in generative grammar).

Logic

As a discipline, logic dates back to Aristotle, who established its fundamental place in philosophy. The study of logic is part of the classical trivium.

Averroes defined logic as "the tool for distinguishing between the true and the false"^[4]; Richard Whately, '"the Science, as well as the Art, of reasoning"; and Frege, "the science of the most general laws of truth". The article Definitions of logic provides citations for these and other definitions.

Logic is often divided into two parts, inductive reasoning and deductive reasoning. The first is drawing general conclusions from specific examples, the second drawing logical conclusions from definitions and axioms. A similar dichotomy, used by Aristotle, is analysis and synthesis. Here the first takes an object of study and examines its component parts, the second considers how parts can be combined to form a whole.
Logic is also studied in argumentation theory.^[5]

Historical Figures Lead Us to the Topic of Entanglement

Einstein Exhibit

We regard quantum mechanics as a complete theory for which the fundamental physical and mathematical hypotheses are no longer susceptible of modification.

--Heisenberg and Max Born, paper delivered to Solvay Congress of 1927

You know I have watched the long drawn out conversation on Backreaction about what was once already debated, to have advanced to current status in the world represented as a logic orientated process with regard to entanglement.

What are it's current status in terms of its expression experimentally to know what it is we are doing with something that had been debated long ago?



Solvay Physics Conference 1927 02:55 - 2 years ago

The most known people who participated in the conference were Ervin Schrodinger, Niels Bohr, Werner Heisenberg, Auguste Piccard, Paul Dirac, Max Born, Wolfgang Pauli, Louis de Broglie, Marie Curie, Hendrik Lorentz, Albert Einstein and others. The film opens with quick shots of Erwin Schrodinger and Niels Bohr. Auguste Piccard of the University of Brussels follows and then the camera re-focuses on Schrodinger and Bohr. Schrodinger who developed wave mechanics never agreed with Bohr on quantum mechanics. Solvay gave Heisenberg an opportunity to discuss his new uncertainty principle theory. Max Born's statistical interpretation of the wave function ended determinism in atomic world. These men - Bohr, Heisenberg, Kramers, Dirac and Born together with Born represent the founding fathers of quantum mechanics. Louis de Broglie wrote his dissertation on the wave nature of matter which Schrodinger used as basis for wave mechanics. Albert Einstein whose famous response to Born's statistical interpretation of wave function was "God does not play dice." Twenty-nine physicists, the main quantum theorists of the day, came together to discuss the topic "Electrons and Photons". Seventeen of the 29 attendees were or became Nobel Prize winners. Following is a "home movie" shot by Irving Langmuir, (the 1932 Nobel Prize winner in chemistry). It captures 2 minutes of an intermission in the proceedings. Twenty-one of the 29 attendees are on the film. --- It's Never too Late to Study: http://www.freesciencelectures.com/ --- Notice: This video is copyright by its respectful owners. The website address on the video does not mean anything. ---

***

The Einstein-Podolsky-Rosen Argument in Quantum Theory

First published Mon May 10, 2004; substantive revision Wed Aug 5, 2009

In the May 15, 1935 issue of Physical Review Albert Einstein co-authored a paper with his two postdoctoral research associates at the Institute for Advanced Study, Boris Podolsky and Nathan Rosen. The article was entitled “Can Quantum Mechanical Description of Physical Reality Be Considered Complete?” (Einstein et al. 1935). Generally referred to as “EPR”, this paper quickly became a centerpiece in the debate over the interpretation of the quantum theory, a debate that continues today. The paper features a striking case where two quantum systems interact in such a way as to link both their spatial coordinates in a certain direction and also their linear momenta (in the same direction). As a result of this “entanglement”, determining either position or momentum for one system would fix (respectively) the position or the momentum of the other. EPR use this case to argue that one cannot maintain both an intuitive condition of local action and the completeness of the quantum description by means of the wave function. This entry describes the argument of that 1935 paper, considers several different versions and reactions, and explores the ongoing significance of the issues they raise.

Might I confuse you then to see that their is nothing mystical about what our emotive states implore, that we might not also consider the purpose of Venn Logic, or, a correlation to Fuzzy logic to prepare the way for how we can become emotive entangled in our psychology, are ways "biologically mixed with our multilevel perspective" about how photons interact, to see that such a color of debate could have amounted to a distinction that arises from within. Which can manifest itself on a real world stage that is psychological forced out of the confines of human emotion, to be presented as a real world force "bridle or unbridled" with regard to the human condition?

See :

Entanglement Interpretation of Black Hole Entropy 

See Also:Backreaction: Testing the foundations of quantum mechanics

Jung Typology Test

Take the Test here.

* Your type formula according to Carl Jung and Isabel Myers-Briggs typology along with the strengths of the preferences
* The description of your personality type
* The list of occupations and educational institutions where you can get relevant degree or training, most suitable for your personality type - Jung Career Indicator™

About 4 Temperaments

So you acquiescence to systemic methods in which to discern your "personality type." You wonder what basis this system sought to demonstrate, by showing the value of these types? So why not look? Which temperament do you belong too?

Idealist Portrait of the Counselor (INFJ)

Counselors have an exceptionally strong desire to contribute to the welfare of others, and find great personal fulfillment interacting with people, nurturing their personal development, guiding them to realize their human potential. Although they are happy working at jobs (such as writing) that require solitude and close attention, Counselors do quite well with individuals or groups of people, provided that the personal interactions are not superficial, and that they find some quiet, private time every now and then to recharge their batteries. Counselors are both kind and positive in their handling of others; they are great listeners and seem naturally interested in helping people with their personal problems. Not usually visible leaders, Counselors prefer to work intensely with those close to them, especially on a one-to-one basis, quietly exerting their influence behind the scenes.

ounselors are scarce, little more than one percent of the population, and can be hard to get to know, since they tend not to share their innermost thoughts or their powerful emotional reactions except with their loved ones. They are highly private people, with an unusually rich, complicated inner life. Friends or colleagues who have known them for years may find sides emerging which come as a surprise. Not that Counselors are flighty or scattered; they value their integrity a great deal, but they have mysterious, intricately woven personalities which sometimes puzzle even them.

Counselors tend to work effectively in organizations. They value staff harmony and make every effort to help an organization run smoothly and pleasantly. They understand and use human systems creatively, and are good at consulting and cooperating with others. As employees or employers, Counselors are concerned with people's feelings and are able to act as a barometer of the feelings within the organization.

Blessed with vivid imaginations, Counselors are often seen as the most poetical of all the types, and in fact they use a lot of poetic imagery in their everyday language. Their great talent for language-both written and spoken-is usually directed toward communicating with people in a personalized way. Counselors are highly intuitive and can recognize another's emotions or intentions - good or evil - even before that person is aware of them. Counselors themselves can seldom tell how they came to read others' feelings so keenly. This extreme sensitivity to others could very well be the basis of the Counselor's remarkable ability to experience a whole array of psychic phenomena.

When you "discover a symbol" as indicated in the wholeness definition presented below, you get to understand how far back we can go in our discoveries. While I talk of Mandalas, I do for a reason. While I talk of the inherent nature of "this pattern" at the very essence of one's being, this then lead me to consider the mathematical relations and geometries that become descriptive of what we may find in nature with regards to the geometric inclinations to a beginning to our universe? How nice?

Wholeness. A state in which consciousness and the unconscious work together in harmony. (See also self.)

Although "wholeness" seems at first sight to be nothing but an abstract idea (like anima and animus), it is nevertheless empirical in so far as it is anticipated by the psyche in the form of spontaneous or autonomous symbols. These are the quaternity or mandala symbols, which occur not only in the dreams of modern people who have never heard of them, but are widely disseminated in the historical records of many peoples and many epochs. Their significance as symbols of unity and totality is amply confirmed by history as well as by empirical psychology.[The Self," ibid., par. 59.]

Update:

See:Expressions of Compartmentalization

Democritus had Passion and Heat?

It seems "humour" is pervading the internet today, so I thought I would add my take.


Democritus Laughing, by Hendrick ter Brugghen, 1628, in Rijksmuseum, Amsterdam

According to legend, Democritus was supposed to be mad because he laughed at everything, and so he was sent to Hippocrates to be cured. Hippocrates pointed out that he was not mad, but, instead, had a happy disposition. That is why Democritus is sometimes called the laughing philosopher

If one had never understood the entanglement process" might one have ever understood what could happen when you mix three circles/sphere of knowledge which overlap to become the "Venn logic of approach?"



Are men suppose to be "Illogical" and "Impassionate?" Maybe "that heat" can refer to the subjective analysis of all the things we might talk about in terms of "creativity?" Yet too, all the things that could involve the human being whilst it engages in the emotive memory induced entrapment of the world inside, which may disallow "clarity of the situation?"

The Art of Doodling

A graph induced analysis of the "boring lecture?" Whose point is the "climatic schedule of the hour," could have ripples following "all the power of that one moment?" While "witnessing this event" the deeper aspect of the student is engaged with things "rising from the unconscious."

Unbeknownst to them, having withdrawn into the dream world, they brought back with them, subjective desires of their soul? Impatience, and "being to the point" while all thing allowed them to journey a long distance from the classroom?

So having drawn this "three circles" or "introducing the "graph of boredom," the idea here is to explain what is "preoccupying the mind" when it should really be paying attention?:)

Democritus, known in antiquity as the ‘laughing philosopher’ because of his emphasis on the value of ‘cheerfulness,’ was one of the two founders of ancient atomist theory. He elaborated a system originated by his teacher Leucippus into a materialist account of the natural world. The atomists held that there are smallest indivisible bodies from which everything else is composed, and that these move about in an infinite void space. Of the ancient materialist accounts of the natural world which did not rely on some kind of teleology or purpose to account for the apparent order and regularity found in the world, atomism was the most influential. Even its chief critic, Aristotle, praised Democritus for arguing from sound considerations appropriate to natural philosophy.
In common with other early ancient theories of living things, Democritus seems to have used the term psychê to refer to that distinctive feature of living things that accounts for their ability to perform their life-functions. According to Aristotle, Democritus regarded the soul as composed of one kind of atom, in particular fire atoms. This seems to have been because of the association of life with heat, and because spherical fire atoms are readily mobile, and the soul is regarded as causing motion. Democritus seems to have considered thought to be caused by physical movements of atoms also. This is sometimes taken as evidence that Democritus denied the survival of a personal soul after death, although the reports are not univocal on this.

Drawing a Venn diagram: Entanglement Issues

Plectics, by Murray Gellman

It is appropriate that plectics refers to entanglement or the lack thereof, since entanglement is a key feature of the way complexity arises out of simplicity, making our subject worth studying.

The person above was kind enough to send information held in context of picture link for consideration, to help out with comprehension. I mean certain things hold us to consequences, that while I might have been thinking of Einsteins example of a pretty girl and a hot stove, this thinking did not pass my attention when one held the photon to certain enviromental influences as we gave these things thought processes. I context of "gravity as the square," is appropriate I think, about what combinations are realistic.



At the very heading of this post there is a link directly attached for consideration in context of all these possibilties. Some things come to mind in terms of Feynman's toy models, as strict interactive phase that we would like to keep track of. So what one might have done to say, hey, if we are given a possibility of scenarios about Entanglement issues, how shall we solve these interactive phases, as we try to build a multiphase integrative model held in context that perfect human being.

Entanglement applies to two or more particles even if one of them is used as input to the two slit experiment, it is not applicable to single particle experiments.

YOu know it is not that simple, but there are always grand designs on what we think something is nifty in society, as we progress our models of the future. As to how we will create the perfect models for apprehension about our universe, and how we interact with it.

Pictorial represenations can be very useful in presenting information or assisting reasoning. Venn diagram is an example. Venn diagrams are used to represent classes of objects, and they can also assist us in reasoning about the relations between these classes. They are named after the English mathematician John Venn (1834 - 1923), who was a fellow at Cambridge University.

While it is true that I am being fascinated by mathematical processes, and how they are used in our visionary quest for understanding, one would have to be a computer to remember all the interactive phases that could have manifested from a situation held in context of a "societal problem." One we might have encountered in our lifetimes.

It's statistical outcome that held to such micromanagement processes, would have been lost on all our minds, if we did not think some science process could have been touted with all these combinations.



Each time I am presented with this thinking, the elementals of the model for apprehension, it always seemed easier to me to just have a look and see what "buddhist principles" are telling us about how we have a hold of our world in all it's realisms. The choices we make, and how we are to conduct ourselves "becoming." Einstein used that term well I think.

So why such association and "combinations", that we have move the thinking here to what was gained in our emotive and abstract thinkings, as productive human beings? To see what a new foundational logic is being developed around our lives. Did we did not readily see the significance of the technologies involved. One had to dig a little deeper I think.

So here is a preview of what entanglement issue has been shown to help orientate views on this issue. Some diagram perhaps, to show the developing scenarios around such entanglement issues?



Quite early these indications about the possibilties of entangled states, raised all kinds of questions in my mind. Thinking of Hooft and others, about the issues of classical quantum processes, over top of these wide and incomprehensible statistical possibilties, seems held under the auspice of our reality model. That "square", given earthbound recognizitons, happily according to the basic pricniples, have so far held our views in gravitational model assumptions. IN essence, we have boxed the views on entanglement. As we have boxed Andrey Kravstov computerized model of the orignations of this universe in a supersymmetrical view of origination. What could have arisen from such situations. Probable outcomes?

Whether such a "quantum computer" can realistically be built with a value of L that is large enough to be of practical use is a topic of much debate. However, the mere possibility has led to an explosive renaissance of interest in the host of curious and classically counterintuitive properties associated with entangled states. Other phenomena that rely on nonlocal entanglement, such as quantum teleportation and various forms of quantum cryptography, have also been demonstrated in the laboratory

Venn Logic and TA

I always lke to inject a piece of my young child's perspective, becuase it helps explain things a bit about "the neighborshood." As we move through them. He asked me about the neighborhood that we were moving too, as if, that was the world. That was his question. "Is this the World," Dad? He was about 4 years old at the time. He's gone now, married, and leadng a very productive life.



The purpose of this picture above comes later down the length of this post. We are all not perfect, are we?

As a youth my first reading of this topic was from a book called, I'm Okay Your Okay, by M.D. Thomas A. Harris.

It helped in defining aspects of myself in context of what was happening in my environment, while being introspective. Is this always a good thing that we take stock of what is going on inside ourselves to wonder indeed what is all these parts of ourselves. How do we look at them in context of the way the "I" is being used.

These lessons were also derived from other places, so it becomes culminative in how one might look at society, and from it, a biased view forms no dfferent then one who has been transplanted from historical context of experience, moving from the lands of the Carribean to the UK society, held in context and views of the community in which they had lived, and live now. There were comparative features drawn from expeirence to consider here?

So as you move, you grow in perspective, all the while the pictures of mountains and similarities strike poises in our mind of of wnating recognition. Who was this person at the time of youth, who, like myself saw farms of a early history, to know that today this historical past is all but forgotten in the real world struggle of lives held in context of these farms? The picture today, contains the picture of our youth.

Parent ego state

This is a set of feelings, thinking and behaviour that we have copied from our parents and significant others.

As we grow up we take in ideas, beliefs, feelings and behaviours from our parents and caretakers. If we live in an extended family then there are more people to learn and take in from. When we do this, it is called introjecting and it is just as if we take in the whole of the care giver. For example, we may notice that we are saying things just as our father, mother, grandmother may have done, even though, consciously, we don't want to. We do this as we have lived with this person so long that we automatically reproduce certain things that were said to us, or treat others as we might have been treated.

As we grow up we take in ideas, beliefs, feelings and behaviours from our parents and caretakers. If we live in an extended family then there are more people to learn and take in from. When we do this, it is called introjecting and it is just as if we take in the whole of the care giver. For example, we may notice that we are saying things just as our father, mother, grandmother may have done, even though, consciously, we don't want to. We do this as we have lived with this person so long that we automatically reproduce certain things that were said to us, or treat others as we might have been treated.

Adult ego state

The Adult ego state is about direct responses to the here and now. We deal with things that are going on today in ways that are not unhealthily influenced by our past.

The Adult ego state is about being spontaneous and aware with the capacity for intimacy. When in our Adult we are able to see people as they are, rather than what we project onto them. We ask for information rather than stay scared and rather than make assumptions. Taking the best from the past and using it appropriately in the present is an integration of the positive aspects of both our Parent and Child ego states. So this can be called the Integrating Adult. Integrating means that we are constantly updating ourselves through our every day experiences and using this to inform us.

In this structural model, the Integrating Adult ego state circle is placed in the middle to show how it needs to orchestrate between the Parent and the Child ego states. For example, the internal Parent ego state may beat up on the internal Child, saying "You are no good, look at what you did wrong again, you are useless". The Child may then respond with "I am no good, look how useless I am, I never get anything right". Many people hardly hear this kind of internal dialogue as it goes on so much they might just believe life is this way. An effective Integrating Adult ego state can intervene between the Parent and Child ego states. This might be done by stating that this kind of parenting is not helpful and asking if it is prepared to learn another way. Alternatively, the Integrating Adult ego state can just stop any negative dialogue and decide to develop another positive Parent ego state perhaps taken in from other people they have met over the years.

Child ego state

The Child ego state is a set of behaviours, thoughts and feelings which are replayed from our own childhood.

Perhaps the boss calls us into his or her office, we may immediately get a churning in our stomach and wonder what we have done wrong. If this were explored we might remember the time the head teacher called us in to tell us off. Of course, not everything in the Child ego state is negative. We might go into someone's house and smell a lovely smell and remember our grandmother's house when we were little, and all the same warm feelings we had at six year's of age may come flooding back.

Both the Parent and Child ego states are constantly being updated. For example, we may meet someone who gives us the permission we needed as a child, and did not get, to be fun and joyous. We may well use that person in our imagination when we are stressed to counteract our old ways of thinking that we must work longer and longer hours to keep up with everything. We might ask ourselves "I wonder what X would say now". Then on hearing the new permissions to relax and take some time out, do just that and then return to the work renewed and ready for the challenge. Subsequently, rather than beating up on ourselves for what we did or did not do, what tends to happen is we automatically start to give ourselves new permissions and take care of ourselves.

Alternatively, we might have had a traumatic experience yesterday which goes into the Child ego state as an archaic memory that hampers our growth. Positive experiences will also go into the Child ego state as archaic memories. The positive experiences can then be drawn on to remind us that positive things do happen.

The process of analysing personality in terms of ego states is called structural analysis. It is important to remember that ego states do not have an existence of their own, they are concepts to enable understanding. Therefore it is important to say "I want some fun" rather than "My Child wants some fun". We may be in our Child ego state when we say this, but saying "I" reminds us to take responsibility for our actions.

Go ahead and Click on John Venn's picture

A painting of John Venn by Charles E. Brock. Photograph by Christopher Hurst, Hamilton-Kerr Institute, University of Cambridge

IN a strange world where such things are to become part of our analysis of subjective experience will we have created the perfect human being? Such models becme the logic structure of new computerized systems of thinking that will somehow recognize these factors in the human beings that we are, to have it done better in a traditional methods of logic, formed around and in computerized thinking.

Science and TA by Chris Boyd

Do we selectively ignore other models from artificial intelligence such as Zadeh's Fuzzy Logic? This is a logic used to model perception and used in newly designed "smart" cameras. Where standard logic must give a true or false value to every proposition, fuzzy logic assigns a certainty value between zero and one to each of the propositions, so that we say a statement is .7 true and .3 false. Is this theory selectively ignored to support our theories?

I had engaged this kind of exercise on my own, as I moved through different research avenues trying to piece together in my own mind, a process held in context of these neighborhoods, and such. How such interactive phases might have been thought about in science procedures. Looking at the basis of this logic, I was looking to find comparsions for this thinking in which the foundational perspectives might make sense.

Science and TA by Chris Boyd

If deduction is another way of knowing in TA, where and when is it applicable? Berne did use Venn diagrams, circles either distinct or overlapping, which were borrowed directly from symbolic logic to visually describe transactions. Perhaps TA in part can be considered a theory that analyzes ones own deductions based on childhood primitive assumptions. Perhaps it focuses on how people become irrational in decision making. In this case, TA provides critical thinking skills for human relations and can be considered a basis for analyzing the accuracy of our reality testing. Much as mathematics provides the language for science, TA may provide the logic for human relations and can be at least in part a deductive language.

Now as usual I did not have many to direct my thinking as I was developing thought processes, that the statements I highlighted were recognized as features I myself found along the journey to make sense of characteristic features we have of our thinking minds.

As you see clicking on John Venn's picture I would like to draw science into how I am seeing to support this "model of thinking." For clear and consise methods incorporating that clear mind.

Do we have to be robots with parts of ourselves missing from the supposed framework of that perfect being? This is what makes us unique in our perspectives as we move forward and share our point of views. We would have had to have been you to become who you are today. So it is never easy to see and be all that you want everyone to be.