Rickert’s Stick-Figure Syllogisms
SUMMARY:
Can we “do logic” with diagrams?
Reason with pictures?
“Yes We Can”.
People have been reasoning (reliably) with pictures for hundreds of years.
But existing systems for traditional logic have drawbacks: some are clunky, others obscure, and one requires a set of assumptions from modern logic.
A better system was sorely needed.
And a Catholic priest, Fr. Rickert, has created one!
It’s simple, elegant, and allows the rapid solving of syllogisms—visually.
Read on to learn more!
Not Picture Perfect
What is given by reason becomes both more teachable and better remembered by us with diagrams and figures…
The main diagrammatic systems in traditional logic are the Byzantine, Euler and the Venn diagram.
I’ll briefly describe each diagram style:
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Byzantine: Originating in the Byzantine (Greek) Empire, sometime between the 4th and 6th centuries ad, as a teaching aid for Aristotle’s logic. They eventually made their way into Europe, and were widely used up until the 16th century.[1]
Byzantine diagrams of the first, second and third figures (left-to-right). They use crescent, circle and triangle shapes to both solve syllogisms and convert between them, and recent academic work has shown they can be extended beyond traditional logic.[2]
Why aren’t they still used? Well, as befits something termed “Byzantine”, they are rather clunky and complicated to use, requiring reduction of all syllogisms to the “perfect” moods of the first figure, rather than solving each case from first principles. People likely just switched to more workable formats, as they became available (e.g. Euler diagrams).
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Euler: Used by the mathematician Leonard Euler in his Lettres à une Princess d’Allemagne, a compilation of letters he wrote (on various educational topics) between 1760 and 1762.[3] Euler was not trying to create a new diagrammatic framework in these letters, but they enjoyed widespread popularity—resulting in his diagrams becoming the default standard in Europe.
All four traditional propositions, as Euler originally depicted them. Euler’s general approach was to depict the terms of propositions as circles, with the position of the circles to each other determined by the type of the proposition. Individual proposition diagrams could be composed together, to show valid syllogisms.
While his approach was simple and intuitive, his diagrams have drawbacks: distinguishing particular from universal propositions is not obvious, existential import is implicitly assumed for denials, and showing invalid forms of reasoning can be ambiguous and cumbersome.
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Venn: In 1880, John Venn, the English mathematician and logician, introduced his revolutionary logical diagrams in his paper On the Diagrammatic and Mechanical Representation of Propositions and Reasonings.[4]
Venn diagram solving a syllogism in mood “Barbara”, using the modern logic proposition Am: “There is no X that is not Y”. Venn’s system represents classes of things directly, using shapes like circles and ellipses, and shows every possible combination by overlapping each shape with all the others. Importantly, we shade an area not to draw attention to it, but to indicate that the compartment (or class) does not exist.
There are several disadvantages to Venn diagrams, the most important being they are founded on modern logic, a system fundamentally incompatible with traditional logic. Furthermore, they quickly become unwieldy—even unusable—as the number of terms increases, and identifying invalid forms requires interpretation from the viewer.
So, for depicting and solving traditional syllogisms, the main diagrammatic systems were lacking.
Enter Father John Rickert, a Catholic priest, who proposed a brilliant and innovative diagrammatic system for traditional logic, in his 2018 paper: A Simplified Approach to Traditional Formal Logic.[5]
Technically, his system is an adaptation of mathematical diagrams called Hasse diagrams, which are used to depict partially-ordered sets (“posets”); it is an ingenious system and personally represents the most exciting development in visual logic since Venn’s work in the late 19th century.
(I’ll call them “Rickert diagrams” from now on)
For this article, I will ignore the origins of Fr. Rickert’s diagrams in set theory and focus on what it provides us with in logic: simple stick-figures that solve syllogisms.
Positive Propositions
Universal Affirmatives
We’ll start with the universal positive (“affirmative”) proposition, traditionally labeled ‘A’:
- A: All S are P
The Rickert diagram is very simple in this case: a vertical line segment with circles on its ends.
We call the end-points “vertices”, and the bottom point (vertex) is labeled “S”, the top “P”.
Circles are placed on the vertices (end-points) when we are speaking about all instances of a category or type of thing.
Later, we’ll see vertices without circles, when talking about some members of a category, rather than all.
This diagram conveys the idea that all of S is contained within P. You could say S is a proper subset of P, S ⊂ P, but I’ll leave set theory out of this (it just complicates things).
The rule for reading these diagrams is to start with the bottom vertex (‘S’ in this case) and work your way up. There’s no going down in these diagrams—only up.
This single direction of travel—only upwards!—leads to one of key benefits of the Rickert diagram: it makes invalid syllogisms immediate and obvious, as we shall see later.
Solving Barbara
Now to actually solve a syllogism using a Rickert diagram.
We’ll use a syllogism in the mood Barbara: first figure, with universal affirmative (positive) premises.
A: All Mammals are Animals & A: All Dogs are Mammals ∴ A: All Dogs are Animals
The symbol ∴ means “therefore”, and is used to indicate the conclusion of a syllogism.
To solve this using a Rickert diagram, we literally just join the premises together on their common middle term (“Mammals”).
The premises give us an ordered relation between Mammals and Animals, and between Dogs and Mammals; thus we can conclude that Dogs are included in the group Animals.
In the symbols of set theory: Dogs ⊂ Mammals ⊂ Animals We established the relation between Dogs and Animals indirectly, using logic, linking these terms via their connection to the middle term “Mammals”.
The Rickert diagram visually captures this notion of connecting concepts, by connecting the matching vertices of the two premise figures, to give us the conclusion.
Particular Affirmatives
Particular affirmative (“positive”) propositions, labeled with the letter ‘I’, are those of the form:
- I: Some S are P
We make a positive assertion (we affirm something) about some part of a category or type of thing: I: Some Dogs are Mesocephalic
Visually, Rickert diagrams depict particular affirmative propositions as a line segment with a “kink”, a sharp turn, in it.
This “kink” between the end-point vertices S and P forms a corner; this corner will also be called a vertex, like the end-points.
But, unlike the end-point vertices, we will not add a circle at this corner vertex.
Why?
Because we only add circles to vertices when we are talking about all members of a category.
When we are only talking about some members of a category, we leave the vertex as is.
We label the corner “S & P”, to indicate that something exists that belongs to both categories, S and P.
Note: “some” means something different in traditional logic, when compared to everyday English.
In traditional logic, “some” means “at least one”, i.e. “not none”, and this can include “all”. This is confusing, since in everyday English, “some” means “not none, but less than all”.
Anyways, just remember: “some” just means “at least one” in traditional logic.
The Rickert diagram clearly distinguishes between universal (all) and particular (some) propositions, allowing easy combination of propositions, and subsequent interpretation—as we will see in the next section.
Solving Darii
Now to solve a syllogism with a particular proposition, using a Rickert diagram.
We’ll use a syllogism in the mood Darii: first figure, with a universal affirmative (positive) major premise, and a particular affirmative minor premise.
A: All Felines are Carnivores & A: Some Mammals are Felines ∴ A: Some Mammals are Carnivores
Depicting the premises only in a Rickert diagram, we get the following, which we will combine together in the next step:
As before, we literally just join the premises together on their common middle term (“F” for “Felines”), rotating the major premise as we connect them.
After connecting the premises, we get a particular conclusion (“some Mammals are Carnivores”), which looks like a checkmark.
Whenever you have a particular conclusion in a Rickert diagram, you will have this checkmark figure: it is one of the two valid forms of a conclusion figure, the other being the vertical line segment we saw when solving Barbara.
Positively Invalid
Given any two categories, S and P, there are five possible relations between them, called Gergonne Relations:
- Identity: They are the same thing, i.e. S is P.
- Inclusion: S includes P, in its entirety.
- Contained: S is contained in P, in its entirety.
- Intersection: there is some overlap between S and P.
- Exclusion: No overlap between S and P, in any way.
The French mathematician Joseph Diaz Gergonne seems to have been the first to publicly describe these relations and apply them to syllogisms.[6]
We can depict these various relations using Euler diagrams:
In traditional syllogisms neither identity nor inclusion are considered, but the other three Gergonne relations apply: contained (A), intersection (I, O) and exclusion (E).
The entire purpose of traditional syllogisms is to establish a definite relation between two terms (the subject, S, and the predicate, P), by means of a connecting “middle” term, M.
A syllogism is called “valid”, when it eliminates the alternative relations, leaving only a single relation between S and P, whatever that may be.
This means the conclusion follows necessarily from the premises, allowing us to be absolutely sure in our reasoning.
No guessing in logic! Only certainty!
It is the structure of a syllogism, the arrangement of its terms, which determines its validity, and Rickert diagrams make this structure explicit.
You can tell immediately if a conclusion is valid, and if it is not, you can easily tell why.
There are only two valid conclusion figures in Rickert’s system: the vertical line segment and the checkmark, which we saw above when solving Barbara and Darii.
These figures are conclusions (rather than premises), since S and P are linked by the “middle” term, M.
Any conclusion that is not one of these two figures is invalid.
That’s it!
(Dealing with denials in Rickert’s system is more complicated, and I have yet to discuss them, but the same rules apply.)
For example, consider two invalid syllogisms of the first figure:
- II: two particular premises (type ‘I’).
- IA: a particular major premise (‘I’) and a universal minor premise (type ‘A’).
The Rickert diagrams for these two invalid syllogisms are visually distinctive, making it obvious that they are invalid.
The II syllogism results in a “W” shape, and since we can only travel “up” in a Rickert diagram, there is no path connecting any part of S to P: we cannot establish any relation between S and P based on the premises.
The IA syllogism gives another invalid shape: like an “M” with the right stem missing. Again, like the “W” shape, we cannot establish a relation between S and P based on the premises.
It is immediately obvious that the shapes resulting from combining these premises are not the same as the valid conclusion figures.
The strikingly clear depiction of invalid conclusions really sets Rickert’s system apart.
Euler’s system requires you to draw out all of the possible alternatives if you want to depict an invalid conclusion, which is cumbersome; Venn diagrams are better in this regard, but recognizing invalid syllogisms requires interpretation and can be subtle.
Negative Propositions
Positive Affirmations, Negative Denials
Aristotle described two general types of propositions:
- Affirmations: You make a positive claim, that something is the case. The ‘A’ and ‘I’ propositions are of this type; their labels come from the first two vowels of the Latin verb affirmo (“I affirm”).
- Denials: You make a negative claim, saying that something is not the case. The ‘E’ and ‘O’ propositions are of this type; their labels come from first two vowels of the Latin verb nego (“I deny”).
This fundamental distinction between affirmations (positive) and denials (negative) has consequences for visualizing the different types of propositions.
Visually affirming something is easy: you just draw it.
Draw two circles to represent two different categories, and you are visually asserting that these categories exist.
But how do you draw a denial?
You are not saying what is, only what is not.
We see one graphical solution in everyday signs, where the general prohibition sign, 🛇︎, is overlaid on things to indicate that something is not permitted.
No smoking. No parking. No left turns, and so on.
Venn solved this issue by shading-out areas to indicate they did not exist, but Euler never settled on a convention for denials and implicitly depicted denials as affirmations.
What do I mean by saying Euler “depicted denials as affirmations”?
I’ll explain in the next section, but I have written more on this point in the footnotes*.
Euler’s Implicit Obversion
Consider Euler’s depiction of the universal negative proposition, ‘E’:
- E: “No S are P”
This is a denial, but what Euler did was depict it visually as an affirmation: that “All S are non-P”, where non-P is everything that is not P.
All of S is clearly in the region non-P (shaded), which is everything that is not P.
To me, Euler’s graphical language is purely affirmative: he makes positive claims about S, visually, whether S belongs to P, or S belongs to non-P.
This is absolutely fine of course, but I just want to make it explicit that we are dealing only with affirmations, since I have never seen this important point emphasized when discussing Euler’s diagrams.
Importantly, “All S are non-P” is not the exact same thing as “No S are P”.
The proposition “No S is P” is true if ‘S’ does not exist, but “All S are non-P” would be false in this case.
But, if we assume that S always exists, then we can happily convert denials (E and O) into their respective affirmative form, called the ‘obverse’, which I will indicate with an asterisk, *:
- E: No S is P
- E*: All S is non-P
And for the particular negative, O, which is typically worded as “Some S is not P”:
- O: Some S is not P
- O*: Some S is non-P
This is the approach that Rickert takes to depict denials in his diagrammatic system: he recasts denials as affirmations. Unlike Euler, Rickert makes this conversion of negative premises (denials) into positive ones (affirmations) explicit in his system.
The British mathematician and logician Augustus De Morgan had developed an affirmative-only logic from at least 1846, showing it equivalent to traditional logic.[7] Alas, this aspect of his work never really went anywhere, so it is exciting to see this novel approach re-appear in Rickert’s system.
Universal Denials
The universal negative (“denial”) proposition, is traditionally labeled ‘E’:
- E: No S are P
As mentioned, we need to convert this denial to its affirmative obverse, E*:
- E*: All S are non-P
Where “non-P” is everything that is not P.
(In set theory, “non-P” is called the compliment, and is symbolized as P-prime: P′)
Once we do this, the Rickert diagram is both simple and familiar: it is the same as the universal positive, ‘A’, with a top vertex of “non-P” rather than “P”.
That’s it!
Everything else is the same as for the vertical line segment, which we saw previously for universal affirmatives.
Connecting Contrapositives
Using only affirmative propositions is wonderfully clever, but it runs into a serious issue rather quickly.
If one premise has non-M as a middle term, and the other M, then you will not be able to connect the two premises together to reach a conclusion.
Consider the following invalid argument, with two negative universal premises, converted to their obverses, E*:
E*: All M are non-P &E*: All S are non-M N/A
As you can see, there is no common middle term between these premises: one is ‘M’, the other ‘non-M’.
This means we can not connect them together, since they are not the same.
We can depict this argument using a Rickert diagram:
What to do here?
For universal affirmative propositions, there is a mode of logical inference called “contraposition”, which I will indicate with a backwards ‘c’, ↄ.
When you take the contrapositive of a proposition, you swap the position of S and P, and negate them both: ‘S’ becomes ‘non-S’ and ‘non-P’ becomes ‘P’, etc.
So, for the universal negative, E, we have the following set of relations:
- E: No S are P
- E*: All S are non-P
- E*ↄ: All P are non-S
Where E*ↄ means “E-obverse-contrapositive”.
Whenever a universal affirmative proposition is true, its contrapositive will be true as well. Same for when such a proposition is false: its contrapositive will also be false.
You can exchange a (universal affirmative) proposition for its contrapositive in any syllogism and the overall logic remains unchanged. That is, a universal affirmative proposition is logically identical with its contrapositive.
Now, returning to our original EE syllogism, we see that we can connect the two premises.
Still, even after connecting the premises, we cannot draw a conclusion here.
E*ↄ: All P are non-M &E*: All S are non-M ∴?
Why?
While S and P are both part of non-M, we do not know how they relate to each other; the premises do not give us enough information to establish a definite relation.
Any of the three Gregonne relations could apply: contained, overlap or exclusion.
The Rickert diagram captures this ambiguity succinctly, with an invalid figure that looks like an ‘A’ without the crossbar.
In short: you need contrapositives when using Rickert’s system, especially when solving syllogistic figures beyond the first, e.g. Camestres, etc.
Solving Celarent
Now, to solve a syllogism involving negative premises.
We’ll take the syllogism in the mood Celarent:
E: No M are P &A: All S are M ∴E: No S are P
As discussed, we need to convert the negative premise and conclusion into their affirmative obverses, to depict them in a Rickert diagram.
This is easy to do: we just change each ‘E’ proposition to its obverse, ‘E*’:
E*: All M are non-P &A: All S are M ∴E*: All S are non-P
Showing this in a Rickert diagram, we get the following:
You will recognize this as having the same form as the Rickert diagram for Barbara above.
Two different syllogistic moods, same visual solution (just with different labels).
I find this simplicity one of the key appeals of the Rickert diagrammatic system.
And things stay simple when we come to the final proposition type, the particular negative, type ‘O’.
Particular Denials
Particular negative propositions, labeled with the letter ‘O’, are denials of the form:
- O: Some S are not P
Personally, I prefer the phrasing “not every S are P”, as it makes it clear that type ‘O’ propositions are denials without existential import, but the “Some S are not P” phrasing is most common.
Rickert’s system treats type ‘O’ propositions exactly like their universal negative brethren: we convert to their affirmative obverses, O*, to draw them with our familiar stick-figures.
- O*: Some S are non-P
Now, rather than a denial, we make a positive assertion about some part of the category ‘S’: that at least one part of ‘S’ belongs to the category ‘non-P’.
Exactly like with particular affirmative propositions, Rickert diagrams depict O* propositions as a line segment with a “kink”, a sharp turn, in it.
So, for particular negatives, we have the same stick-figure as particular positives, which we will combine in the same way when solving syllogisms with particular premises.
Solving Ferio
Now to solve a syllogism with a particular negative proposition, type ‘O’, using a Rickert diagram.
We’ll use a syllogism in the mood Ferio: first figure, with a universal negative (denial) major premise, and a particular affirmative minor premise:
E: No M is P &I: Some S is M ∴O: Some S is not P
Of course, as you can probably guess at this stage, we need to convert the negative propositions to their affirmative obverses, obtaining the following syllogistic format:
E*: All M are non-P &I: Some S are M ∴O*: Some S are non-P
Here are the premises, which we will combine together in the next step:
As before, we literally just join the premises together on their common middle term (M), rotating the major premise (E*) as we connect them.
After connecting the premises, we get a particular conclusion, O* (“some S are non-P”), which looks like a checkmark, like with Darii, above.
Once again, Rickert’s system provides a simple visual solving of a syllogism, using the same stick-figures as before, just with different labels (“non-P” rather than “P”).
This continuous re-use of the same simple shapes, across all syllogistic forms—valid and invalid—is what makes this system so powerful and fun to use.
Studying syllogisms can be uninspiring, requiring a lot of memorization and rote mechanisms; Fr. Rickert’s stick figures bring an ingenious and lively approach to the subject, and I look forward to using them going forward.∎
Footnotes
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Personally, if you want to draw a denial of something, you should mark it with an ‘❌︎’ through it, or the general prohibition sign, 🛇︎, to show that it is not the case.
Now, ‘❌︎’ is used to “mark the spot” where literary pirates bury their treasure, and John Venn used it to indicate a region was non-empty in his diagrams, i.e. that something exists (a particular affirmation).
So, clearly, different conventions exist; but to me, an ‘❌︎’ means “no” or “not the case”.
Key Takeaways
- Existing visual systems have significant drawbacks for traditional logic: Venn diagrams are based on modern logic; Euler’s better suited to depicting propositions, rather than solving syllogisms.
- Fr. Rickert’s new system is fast, simple and makes the solving of traditional syllogisms vivid and intuitive.
- The system is derived from Hasse diagrams (which depict partially ordered sets, “posets”), and uses stick-like figures to model traditional syllogisms.
- Obversion, a mode of logical inference, is used to transform all denials (E, O) into affirmative-style propositions (E*, O*), allowing a consistent visual depiction.
- Contraposition, another mode of logical inference, is sometimes required to switch and negate the terms of a proposition—ensuring the same connecting term (M or non-M) between premises.
- Fr. Rickert’s system allows the solving of any traditional syllogism from a limited set of basic principles, rather than from rote memorization.
References
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Safran, L. 2020. A Prolegomenon to Byzantine Diagrams. In: The Visualization of Knowledge in Medieval and Early Modern Europe, pp 361–382. Turnhout: Brepols.
Available at https://www.academia.edu/44038009/….
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Lemanski, J. , Bhattacharjee, R. 2024. On the Expressivity of Byzantine Diagrams in Logic. In: Diagrammatic Representation and Inference [Conference], Lecture Notes in Computer Science, Vol. 14981, pp 429–445. Cham: Springer.
Available at https://link.springer.com/chapter/….
(Accessed: 8th June 2025). [Back] -
Euler, L. 1761. Lettre CII. Lettres à une Princesse d'Allemagne sur Divers Sujets de Physique & de Philosophie. St. Petersburg: Printing House of the Imperial Academy of Sciences.
Available at https://scholarlycommons.pacific.edu/….
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Venn, J. 1880. On the Diagrammatic and Mechanical Representation of Propositions and Reasonings. Philosophical Magazine, Series 5, Vol. 10, No. 59, pp. 1–18.
Available at: https://www.cis.upenn.edu/…
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Rickert, J. 2018. A Simplified Approach to Traditional Formal Logic.
Available at: https://www.academia.edu/37584401/…
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Gergonne, J. D. 1817. Essai de Dialectique Rationnelle. Annales de Mathématiques Pures et Appliquées, Vol. 7, pp. 189–228.
Available at: https://www.numdam.org/article/…
(Accessed: 13th June 2025). [Back] -
De Morgan, A. 1846. On the Structure of the Syllogism, and on the Application of the Theory of Probabilities to Questions of Argument and Authority. Transactions of the Cambridge Philosophical Society. Vol. 8, pp. 379–408.
Available at: https://babel.hathitrust.org/…
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