A Brief Introduction to the Alpha Existential Graph (EG) Tutor
EXISTENTIAL GRAPHS (EG) is an efficient and powerful system of logic invented in the 1890s by the American scientist Charles Sanders Peirce. It is easy to learn, particularly if we approach it initially as a game. After you develop the skills of manipulating these diagrams, you will quickly learn how those skills apply to logic.
Here is what a problem in EG looks like. Soon you will be able to solve problems like this:
The SHEET OF ASSERTION
The game of EG is played on a limitless surface called the SHEET OF ASSERTION (SA). Naturally, we can only see a part of it at any one time.
The Sheet of Assertion we will use is a blank portion of the computer screen.
Instead of looking at the sheet of assertion from the side, we want to imagine that we are looking at it from the bottom of an airplane as we fly directly over it.
CUTS AND LEVELS
Upon the Sheet of Assertion we may place two kinds of items — individual capital letters or CUTS. Everyone knows what a capital letter is, so perhaps you are wondering what a cut looks like.
Here we are seeing a cut in three perspectives — a side view, a three-quarter view, and a top view. We shall use the top view because it is easiest to draw on paper. When you see a cut drawn in top view, remember that it has thickness.
The top surface of a cut is called its LEVEL, while its under surface is called its BASE. Notice that the base of this cut rests on the level of the Sheet of Assertion. The Sheet of Assertion (SA for short) has no base, only its top level.
In order to keep track of levels, we will assign numbers to them. We begin with SA, which has a level assigned the number 2. There are no levels numbered below 2, but there can be as many as we need above 2. There is nothing special about labeling the level of SA as 2 — as long as we count sequentially, any number for SA would work, as you will see.
The level (top) of a cut placed on SA is 3 — and the level (top) of a cut placed on THAT cut is 4, and so on.
LEVELS AND CUTS TOGETHER
Here is the letter A placed on SA (level 2):
Let's add a new cut which allows us to place letter B on level 3. It is drawn from the top, as we will routinely do:
THE LEVEL OF A LETTER
If I placed the letter C on a smaller cut, and then placed that that smaller cut on the big cut, on what level would the C be?
What is the level of the letter C?
THE LEVEL OF A CUT
Sometimes we will want to know on which level a cut rests. In our example above, what is the level on which the graph cut-C is resting? Hint — the level on which a cut rests is the level which the base of the cut contacts.
THE LEVEL OF A GRAPH
Now we have enough behind us to understand what a GRAPH is.
A graph is any of the following:
any portion of SA
any letter or combination of letters that do not overlap, overhang, or cover-up each other
any cut or combination of cuts that do not overlap, overhang or cover-up each other
any combination of letters and cuts free of overlaps and overhangs and cover-ups.
Sometimes it is useful to talk about only a smaller piece of a graph. We will call any smaller piece of a graph a SUBGRAPH.
Let’s practice for a moment in order to learn more about the notion of levels. What is the level of D in this graph?
ANALYSIS OF A COMPLEX GRAPH
Here is a complete analysis of the levels involved in this complex graph. Review all instances to be sure you understand each one.
cut-B on 4
cut-D on 4
F on 4
A on 3
cut-A on 2
C on 3
B on 5
D on 5
THE GAME OF EG
To play a game with Existential Graphs, one considers a problem posed by someone whom we will call the GRAPH PRESENTER. The person who solves the problem will be known as the GRAPH EVALUATOR. The problem will be presented using a format seen in our first example.
In this example, which is typical of the way in which problems are presented in our system, everything to the left of the question mark is known as the INITIAL GRAPH, while everything to the right of the question mark is called the GOAL GRAPH.
The problem is whether the Initial Graph can be correctly transformed into the Goal Graph.
Transformations must take place according to five rules. We will now begin to learn them. They are easily learned and easily remembered.
THE DOUBLE CUT RULE
The first rule we shall learn is known as DOUBLE CUT (which we shall abbreviate as DC). The rule states that a double cut pattern may be removed from under any graph, or placed under any graph. What is the double cut pattern, you ask? Well, it looks like this:
In the above graph, * stands for any well-formed graph whatsoever.
Notice some essential features of the double cut pattern. It must have a completely empty zone between the two nested cuts — no letters or other cuts should be present there.
If you are adding a double cut under a graph, imagine yourself picking up the existing graph with one hand, sliding a double cut stack under it, then placing the existing graph on the top.
If you are removing a double cut, think of the transformation as if you were to hold *, the graph on top, in one hand, and with the other hand, remove the two cuts in the double cut pattern, then release the graph to let it fall to whatever level still exists.
Now let’s look at an example of double cut removal:
Notice that the initial graph has a double cut stack under the leftmost R. By means of the double cut rule (DC) we may transform the Initial graph as specified by DC. Here is the result:
Hope you get the idea — to put a double cut under R would be just the reverse. Let’s see that happen:
Now you know one rule. In just a few seconds, you will know two more, Erase and Insert. These two rules are simple to state.
ERASE AND INSERT RULES
The ERASE transformation rule (abbreviated as E) states that any graph resting on an EVEN LEVEL may be removed. The INSERT rule (abbreviated IN) states that any graph of your choosing may be placed upon an unoccupied place upon any ODD LEVEL. Notice one cannot use the Erase rule on odd levels, and one cannot use the Insert rule on even levels.
Let’s take a look at Erase using an example used earlier:
Remember how to count levels? It is an important skill in using both the Erase and Insert rules. In the case of Erase, we must be able to recognize even levels. Here is a review of levels in this example graph:
cut-B on 4
cut-D on 4
F on 4
A on 3
cut-A on 2
C on 3
B on 5
D on 5
Notice that cut-B is on level 4 — thus, by means of the Erase rule, we may transform the present graph by taking taking out cut-B. Let’s do it:
There is an important difference between Erase and Double Cut. Some persons are tempted to want to remove just the cut and leave the B, but remember that in using Erase, if a cut can be removed because its base rests on an even level, anything on top of the cut being removed goes too.
Now let’s remove F using Erase:
Notice that the entire subgraph we have been working within is itself resting on a level, and that level (the sheet of assertion) is an even level. Thus we may use the Erase rule to remove that entire subgraph — it would look like this:
That is how easy Erase is.
THE INSERT RULE
Now for INSERT. It is even simpler — the rule states that ANY well-formed graph of our choosing may be placed into an unoccupied odd level, provided that we don’t cover anything already there.
If we wanted to place the graph H on level three of our current example, it could be placed in an open area, like this:
Or, suppose we wanted to place a cut-L on level three — it could be done by insertion like this:
That's all there is to it.
ITERATE AND DEITERATE RULES
Next you will learn the last two of the five rules for Existential Graphs. The remaining two are ITERATE (abbreviated as IT) and DEITERATE (abbreviated as DE).
To illustrate the Iterate rule, let's start with the following graph:
IT is rather a simple rule, stating that any graph, either simple or complex,. may be copied or repeated (iterated) within its own CONTINUOUS level as many times as one wishes.
Furthermore, the Iterate rule permits one to make a copy of a graph and place it on some higher level, provided no VALLEYS are crossed.
In viewing the above area, we have the simple graph G written. According to the first aspect of the IT rule, we could repeat this graph as many times as we wished within its continuous level (which, in this case, is the Sheet of Assertion). So let’s repeat it....
We could have repeated G once or as many times as we wished. We can also do this within any other continuous level. For instance, suppose we had a large cut with a G on it...
Here we find a single G lying on level three. We can repeat this once or many times if we wish:
To consider the second part of IT, look at this example:
Suppose that for some reason we would like to repeat cut-A in level four between the B and cut-C. This is a legal transformation, which looks like this:
Suppose instead that we wanted to Iterate just the first A, like so:
That iteration would be illegal because it violates the valley clause of the Iterate rule. You will learn more about valleys on the next page.
Now to consider the Deiterate rule.
The DEITERATE rule permits us to remove one or more duplicated graphs within a single continuous level. For example, if we had a lot of G’s on the same level, we could remove one of them or all of them except for one. Here are a bunch of G’s:
Let’s DE all but one:
Now to consider the second part of DE. This example will help us to understand how it works.
The second aspect of the DE rule states that we may remove a duplication of a graph if the duplicate is found on a higher level than the original and if no valleys are crossed.
In this graph, the cut-A that rests between B and cut-C is a duplicate of the cut-A to the left, which in this case is the original. The cut-A resting on the higher level may be removed with DE.
You can recognize the original because it is always the lower resting graph. The higher resting of the identical graphs is always the one to be removed. It is a false use of DE to remove the lower resting of the identical graphs, leaving the higher.
In this example, the right side cut-A is the one that rests higher, so only it can be removed with DE. Let’s do it:
Now to consider the matter of valleys, a factor common to both DE and IT.
Here is a helpful example with notes.
Suppose we wanted to try to Iterate the C from the place it rests on the left, and place a copy of it just to the left of D on level three. This would not be permitted because to proceed from the original C to the location of its proposed copy would involve crossing a valley.
A valley is involved because to walk from the place of the original C to the place of the proposed copy, one would have to get off level three at the left, descend to level two, and climb back to level three on the right — that’s a valley!
The valley restriction holds also for DE. If you can imagine that there were a C just to the left of D on level three, and you wanted to remove it by DE, it would not be permitted, again, because there is a valley between the duplicates or identical graphs.
This concludes the Introduction to the basic rules and terminology of Existential Graphs.
For an expanded account of EG, see the textbook Elements of Logic (Ketner 1990/1996).