Traditional and Modern Logic
SUMMARY:
Logic is not about being cold and unemotional, like some sort of tin man.
It’s about reasoning: the mental act of going from what we know—to some new knowledge, via indirect means.
We figure things out, rather than having to test them out and learn directly.
It might seem that there would be a unified science of logic, one that was timeless and objective.
But, this is not the case.
Logic is, after all, a human activity.
And humans have different schools of thought.
And two major schools are: traditional logic, derived from Aristotle, and modern logic, developed in the 19^{th} century.
Some claim the modern logic has made the traditional Western logic redundant, viewing it as an almost embarrassing reminder of a backward, medieval age.
Provincial traditional logic versus cosmopolitan modern logic.
Nonsense.
I’ll show these two systems of logic are just different, being grounded on fundamentally different ideas about the world.
Both have their place—and their limitations.
What is “Logic” Exactly?
Logic is the science of reasoning, where reasoning is the mental act of:
 going from what is known (the premises),
 to some new knowledge (the conclusion)
via indirect means.
Reasoning is indirect since we achieve this new knowledge (the conclusion) without direct experience
From what is known, we infer some new knowledge.
As opposed to having to test something out, in the real world, and learn via direct experience.
A simple example: dropping things on your big toe.
Drop a pea on your big toe, and you’ll barely notice.
(Unless you are some sort of princess)
Next, drop a golf ball on your big toe, and it’ll hurt a little.
Now, if after dropping the golf ball, someone said “Let’s drop a bowling ball!”, would you agree?
No, you wouldn’t.
(Unless you’re a masochist or a moron, of course)
You do not need to drop the bowling ball on your foot to figure out that it would hurt a lot—and probably splatter your big toe into a bloody pulp.
In this case, the premises are:
 “A pea does not hurt”, and
 “A golf ball hurts”,
And the conclusion is:
 “A bowling ball hurts a lot / smashes my toe”
The point is that we inferred that a bowling ball would hurt a lot—we didn’t need to drop one to directly learn this.
We figured it out.
And this is the heart of the reasoning process: the ability to gain new knowledge via indirect means.
Logic, as a field of study, was born in Ancient Greece and was developed from the everyday patterns of thinking of people—whether they reasoned correctly or not.
Logic has developed in many directions since, evolving to both study and create patterns of inference more generally.
Under what circumstances will our inferences be justified and thus our reasoning sound?
Logic provides the answers to these questions.
It is the science of inferential structures, of how we can legitimately move from one set of facts to another set of facts, either in the real world or within a formal system.
And by “formal system”, I mean a set of symbols and rules for combining those symbols.
Everyday arithmetic is a great example of a formal system, where:
 The symbols are the familiar numerals (0,1,2,3,4,5,6,7,8,9) and
 The rules are the standard rules of arithmetical operations, e.g. “1 + 1 = 2”, “3 × 4 = 12” etc.
A critical part of reasoning is consistency: the same situation produces the same result.
Always.
If we have a formal system that is consistent within itself, we can say that it is logical.
That is, within the rules of the formal system, we can make inferential steps from the things we know, to new knowledge, confident that we are correct in our reasoning.
The rules are consistent, they do not change and they guarantee their results.
If a system is not logical, then it is illogical, which means it is inconsistent: up can be down, black can be white, your aunt your uncle, and so on.
I’ve mentioned that there are different schools of logic: that people disagree with one another on what sort of inferences should be allowed.
A system (or school) of logic is a given collection of logical rules that are consistent when taken together.
Then, people use a system of logic to make inferences in a formal system.
You’ll see mathematicians use a technique called “proof by contradiction”.
Perhaps you encountered this technique in geometry class and were left befuddled by it.
(I certainly was!)
Perhaps, like me, you wondered why anyone would prove something by first assuming its exact opposite to be true, and then deriving a contradiction, which proved the original thing they wanted to prove.
But is a valid approach (in most schools of logic!) used since Ancient Greece to prove theorems in mathematics, i.e. to confirm new knowledge in a formal system.
In fact, the rules relating a set of symbols to each other in any formal system are just inferential steps taken from a given system of logic.
And a “system of logic” is just a collection of inferential rules, and people are free to create their own.
Of course, your creation may or may not agree with the system, or school, of logic created by others.
This does not mean your system of logic is right and the others are wrong, merely that they are different.
This point is critical: different logical rules give different logical results.
The Logic of Aristotle
No discussion of logic could fail to mention Aristotle and his syllogisms.
Born in Stagirus, Greece in 384 BC, he studied under Plato and was appointed by King Philip of Macedon to tutor his son, Alexander.
(Yes, that Alexander: “Alexander the Great”)
Among many other achievements, he was the first to attempt a systematization of logic, to describe and study the general patterns of inference, independent of the content of arguments.
From this he developed his syllogisms, logical structures that could be used to reason definitely from what was known (the premises) to the previously unknown (the conclusion).
Alas, nearly all of his published works are lost—what precious few works we have are lecture materials, notes and assorted memoranda.
Aristotle’s logic is a logic of classification, of fitting things into the correct categories given some initial information.
Nowadays, we call it a “term logic”, since it deals with the relationship between terms, where a term is a word or phrase that stands for a concept, like “man”, “horse” or “flying saucer”.
Terms are neither true nor false by themselves. We could have a term like “Wakalixes” for instance, which refers to nothing in the current world, and we would call it an empty term, rather than a false term.
A judgement is a comparison between terms, which the mind makes, and establishes some relation between the terms themselves.
Since judgements happen inside our heads, they must be publicly expressed—in words, writing or whatever—before they can be assessed.
When judgements are conveyed to the external world, they are called propositions: a publicly expressed association of terms, e.g. “All Dogs are Mammals”.
As mentioned, Aristotle’s logic concerns itself with terms and their relation to each other—rather than taking propositions themselves as the objects of study (as the logic of the Stoics did).
And it is propositions that are either true or false:
 True when the proposition claims something about the world, which is actually the case.
 False when the proposition claims something about the world, which is not actually the case.
For example, given the terms “Dogs” and “Mammals”, we form the following proposition: “All Dogs are Mammals”
This proposition is true, since all dogs are mammals according to our modern scientific classification.
We can depict this proposition visually, as a circle labelled “Mammals”, fully containing another circle labelled “Dogs”—showing how the entire class of things we call “dogs” belong to the class of things we call “mammals”.
Aristotle’s syllogisms allow no inferences when any of the premises are false: syllogisms work only with premises that are true in the specific aspect of reality you are referring to, which is sometimes called the “universe of discourse” (to use a technical phrase).
So, if you are referring to the real world, but making claims about mythical creatures, then all your propositions will be false, even if they are ostensibly true, e.g. “Some Unicorns are White”.
(Since in European mythology, unicorns are frequently depicted as being white)
However, the ingenuity of his system allows it to be applied to other “worlds”: you just need be clear what world you are referring to.
Aristotle described four fundamental types of propositions, depending on their:
 Quality, either positive (an affirmation, “is”) or negative (a denial, “is not”).
 Quantity, either universal (“all”) or particular (“some” or “at least one”).
Note, “some” in logical usage differs from the every day usage of the word “some”, since logical some may include cases where “all” applies. It helps to think of logical “some” as “not none” or “at least one exists”.
Anyways, the four types of propositions, and their traditional letter labels, are:
 A (universal positive): All Dogs are Mammals.
 I (particular positive): Some Dogs are Mammals.
 E (universal negative): No Dogs are Mammals.
 O (particular negative): Some Dogs are not Mammals.
Aristotle’s genius was in creating structures that combined propositions together in specific ways, which guaranteed correct conclusions, regardless of the content of the terms.
These structures are his syllogisms.
A simple example would be, given the following two pieces of information (the premises): A: “All Mammals are Eukaryotes”, A: “All Dogs are Mammals”, Then, we can conclude: A: “All Dogs are Eukaryotes”.
The brilliance of this approach is that we don’t need to know what a Eukaryote is—or anything else, for that matter.
All that matters is the relationships between terms, whatever those terms may be.
And the rules of Aristotle’s logic still apply to the act of classification today.
They have not been superseded, replaced or made obsolete by more modern developments—as some might claim.
Traditional Logic
Most of what we refer to as “traditional logic” comes not from Aristotle—since most of his works are lost—but from those who came long after him, primarily the Scholastics of medieval Europe.
While the wellspring was Aristotle, it was their tireless work that brought down to the present day, the body of knowledge we call “traditional logic”.
Scholars such as
 Anicius Boethius (c. 480–524 AD),
 Peter Abelard (c. 1079–1142 AD),
 William of Ockham, of “Ockham’s Razor” fame, (c. 1285–1347 AD), and
 Jean Buridan (c. 1300–1358 AD)
performed invaluable work over the centuries and collectively they:
 analyzed, deduced and catalogued the valid and invalid forms of syllogisms,
 devised rules for conversions between syllogistic forms, and
 created diagrams showing the relationship between propositions,
as well as crafting clever mnemonics to easily recall their work.
(Since remembering information was critical in the ages before both the printing press and the internet)
And it was enough, their work—for a while.
The Square of Opposition
The Square of Opposition is a visual depiction of the inferential relations between the four fundamental propositions of Aristotle.
And by “inferential relations” I mean what we can figure out about all the other propositions, from knowing something only about one of them.
If one is true, does it mean the others are true or false?
Of course, you could argue that it should be called the Square of Inference, and I can see your point.
Especially since some of the propositions aren’t exactly opposed to one another—in the modern meaning of the word—since they can both be true at the same time (A and I, E and O, and I and O).
But, this is an argument for another day.
(I think of “opposition” in the sense of “set or placed against”, from the Latin oppositus)
Anyways, given there are four categorical propositions, arranging them into a square seems obvious does it not?
After all, a square has four corners.
But how to arrange them?
In clockwise alphabetical order: A–E–I–O?
Or reverse alphabetical order: O–I–E–A?
Is there a logical arrangement or would it just depend on personal taste?
Unsurprisingly, since this is logic we are discussing, there exists a logical arrangement—one that has been known since Aristotle.
It goes clockwise from the top left corner: A, E, O and I.
This arrangement is based on the following immediate relations between propositions:
 Contradictory (A–O, E–I): both negate the other, and one must be true, i.e. “one or the other.”
 Contrary (A–E): both negate the other, but neither must be true, i.e. “not both.”
 Subcontrary (I–O): both allow the other, and one must be true, i.e. “at least one.”

Altern (A–I, E–O): both allow each other, but neither must be true. Has two forms with opposite directions:
 Subaltern: if the universal is true, the particular is true.
 Superaltern: if the particlar is false, the universal is false.
The terms themselves are derived from Latin, and do not correspond exactly to modern day English, which can make them a little confusing.
For example, “contrary” is often used in the sense of “exact opposite” in English, rather than “opposite quality”, which is how contrary is used in the above set of relations: since A and E are both universal propositions, just one is positive (A) in quality and the other negative (E).
Apuleius of Madaura, writing in Latin during the 2^{nd} century AD, used incongruae (“incongruent”) for contraries and alterutrae (“alternates”) for contradiction, but his phrasing did not catch on.^{[1]}
Though Aristotle discussed these relations—at least implicitly, in the case of subalternation—he did not invent the heuristic diagram, traditionally known as the Square of Opposition, which maps these relations to each other.
And this is what Boethius did, in his work De Syllogismis Categoricis, written c. 505 AD.^{[2]}
He has been called the first of the Scholastics
, and he is largely responsible for conveying Aristotle’s logic through to the medieval period.^{[3]}
We are accustomed to creating diagrams nowadays, living as we do in an age of unrelenting imagery, so drawing a picture might seem obvious, even trivial.
But it was a stunning innovation at the time to consider propositions as abstract entities and place them in a geometric relation to each other.
Boethius built on the work of others in this regard, particularly Apuleius of Madaura, but to Boethius belongs the first published record we have, with the fullyfledged figure—showing all the relations between propositions listed above.^{[4]}
Now, Boethius made an important misstep in phrasing the type O proposition (particular negative) as Some Men are not Just
(Latin: Quidam homo justus non est) in his De Syllogismis Categoricis,^{[5]} rather than “Not all Men are Just”, as per Aristotle’s original phrasing (in Greek).
Curiously, when directly translating Aristotle’s On Interpretation (Greek : Peri Hermeneias) into Latin (in De Interpretatione), Boethius retains Aristotle’s phrasing of “Not every man is white” (Non omnis homo albus est) to illustrate the particular negative.^{[6]}
Apuleius of Madaura, writing several hundred years before, did the same with his phrasing of Some pleasure is not good
(Latin: Quaedam voluptas bonum non est).^{[2]}
While this might seem an irrelevant difference, it has caused significant confusion down the centuries regarding something called existential import.
But I will not dwell here on this error nor on existential import, since I want to emphasize what is ingenious about the Square of Opposition.
Namely, that it depicts fundamental relations between categorical propositions, giving us a visual summary of the system of traditional logic.
Modern logic is incompatible with the traditional logic: in comparing their respective Squares of Opposition (which I will do later), we have an immediate, visual demonstration of these fundamental differences.
A Rhyme to Remember
A “barbarous and wholly unscientific character” no less!^{[7]}
Harsh words indeed from Mr. Jevons.
(Emphasis added to the original)
Why so harsh?
Well, the developers of modern logic—Jevons being one of them—wanted to model the subject after mathematics, and how proofs were conducted in that subject.
Rather than a list of logical results, they wanted to derive all results from a few general principles (axioms) and rules of inference—rather than look up some catalogue of readymade results.
Imagine you wanted to study arithmetic but were told that the subject consisted entirely of memorizing various lists—like the “times tables” we were all harassed into learning as schoolchildren.
You’d think, what a crock!
Surely there is more to the study of numbers (arithmetic) than the memorization of tables of results?
And yes, there most certainly is more to arithmetic than mere memorization: we are taught basic principles, which allow us to tackle an indefinite variety of numerical challenges.
And likewise with logic.
Why should the subject of logic rely on memorizing the following doggerel verse, however useful?
Barbara, Celarent, Darii, Ferioque, prioris;
Cesare, Camestres, Festino, Baroko, secundae;
Tertia, Darapti, Disamis, Datisi Felapton,
Bokardo, Ferison, habet; Quarta insuper addit
Bramantip, Camenes, Dimaris, Fesapo, Fresison.^{[7a]}
While these lines—the venerable “Barbara Celerant”—were certainly clever and useful, Jevons declared that they are altogether an anachronism in the present age.
^{[8]}
(You can read the footnotes* for more information on the meaning of these terms)
Rather than some cumbersome mnemonic rules covering specific situations, those who developed modern logic wanted to analyze any possible form of reasoning—using only a few basic principles.
Modern Logic
For two thousand years, Aristotle’s logic was the central focus of Western logic.
But the times, they changed.
And few places were changing like Europe during the 19th century.
The Industrial Revolution, an age of rapid industrialization and mass mechanization, was transforming the continent and soon, the world of conceptions.
As it forcibly remodeled the landscape with factories, smokestacks and infernal engines, it was to uproot the steady, staid world of logic.
As Man’s strength had been exceeded by the machines of the age, so too would Man’s intellect be exceeded.
The drive was on for algorithmic approaches to intellectual tasks, steps that could be done automatically, mechanistically: a logic of symbols processed mechanically, objectively, flawlessly—that was the goal, the model, the ideal of the age.
A great flurry of work from the late 1840s onwards gave rise to a new logical system, one which displaced Aristotle’s system.
This new logic is called modern logic and it differs from Aristotle’s logic in important respects.
A number of men contributed to modern logic: Boole, de Morgan, Jevons, Venn, Frege, Whitehead, Russell, to name but a handful.
Their work stretched over many years, from (roughly) the 1840s–1930s, and is beyond the scope of this article.
For now, the most important difference between the two systems concerns the existential import of propositions.
Existential Import
What is existential import?
It sounds fancy, but all it means is this: does a proposition assume its terms exist?
When a proposition has existential import, it means the proposition assumes its terms refer to things that exist in the world we are dealing with, the socalled “universe of discourse”.
In Aristotelian logic, If I say “All Greeks are Europeans” then is it implied that both terms (“Greeks” and “Europeans”) refer to things that exist in the real world.
If either “Greeks” or “Europeans” do not exist as things in the real world, then the proposition is false.
For Aristotle, all positive (affirmative) propositions have existential import, but negative propositions (denials) do not.
So, the positive propositions:
 A (“All Greeks are Europeans”), and
 I (“Some Mammals are Aquatic”)
imply the existence of their terms (Greeks, Europeans, mammals and aquatic things).
These positive propositions (A and I) can only be true if their terms refer to things that exist in the world.
Conversely, the negative propositions:
 E (“No Unicorns are Purple”) and
 O (“Some Centaurs are not Surly”)
do not imply the existence of their terms.
When you are denying that something has a certain property, you are correct if the thing in question does not exist.
For things that do not exist, we cannot make any positive claims.
In Aristotle’s logic, negative propositions (E and O) are true when their terms are “empty”, i.e. do not refer to things in the “world” in question.
So, for the real world, we can only make denials about unicorns and centaurs, e.g. “No Centaurs are Mammals”.
(Of course, in the world of European mythology, we could make positive claims, e.g. “Some Unicorns are White”)
Modern logic takes a very different approach to existential import, since modern logic is based on a very different conception of propositions.
Rather than classifying propositions by their quality (positive/negative) and quantity (universal/particular), modern logic recasts all propositions as either affirming or denying existence.
This recasting has been called the Brentano–Venn interpretation, and I will discuss it next.
Brentano–Venn Interpretation
In the traditional system, existential import is determined by the quality of the proposition:
 Positive propositions (A and I) have existential import
 Negative propositions (E and O) do not have existential import
Franz Brentano, a German philosopher and psychologist, in his 1874 work Psychologie vom empirischen Standpunkt, reduced the traditional categorical propositions to propositions either affirming or denying existence.^{[9]}
Rather than the quality (positive or negative) of the proposition determining existential import, Brentano made the quantity (universal or particular) of the proposition the determining factor.
In Brentano’s system:
 Universal propositions (A and E) do not have existential import.
 Particular propositions (I and O) do have existential import.
Arthur Prior called this approach the Brentano–Venn interpretation,^{[10]} and it is at the heart of “modern” logic.
Rather than the traditional proposition, A: “All Unicorns are White”
Modern logic has the proposition, A_{m}: “There does not exist a Unicorn who is not White”
Bit of weird way of phrasing things, isn’t it?
This proposition, A_{m}, is claiming that there is not a single unicorn in the class of all things that are not white.
And, in the real world, this claim is true: there are no unicorns, white or otherwise!
This is in direct contrast to the traditional proposition A (“All Unicorns are White”), which is false in the real world.
(Since the term “Unicorn” is empty, as unicorns are not realworld things)
And so on for the rest of the propositions.
 I (“Some Unicorns are White”) becomes I_{m}: “There exists a Unicorn that is White”
 E (“No Unicorns are White”) becomes E_{m}: “There does not exist a Unicorn that is White”
 O (“Some Unicorns are not White”) becomes O_{m}: “There exists a Unicorn that is not White”
This recasting of the traditional propositions was controversial, one which broke with centuries of logical tradition and created a new system that was incompatible with the old.
One critic, Jan P. N. Land, wrote in 1876 that Brentano’s system …will have the disadvantage which we least expect from an empirical psychologist, of trying to replace a more natural theory by an artificial one.
^{[11]}
Another, even harsher critic, George H. Joyce, wrote: Such a theory carries its own refutation with it. It is manifest to any one who reflects, that as a matter of fact we do not think in these forms.
^{[12]}
Regardless of the objections against it, Brentano’s approach soon became one of the cornerstones of this new logic.
Why?
Well, as per Arthur Prior, the Brentano–Venn interpretation allows complicated inferences to be made with greater ease and elegance
,^{[9]} compared to the traditional interpretation.
By 1880, Brentano’s approach was already being popularized: John Venn, the English mathematician and logician, published a paper introducing his (soon to be famous) logical diagrams.^{[13]}
In it, he states his diagrams are to be constructed in the following manner:
As we can clearly see, Venn is using the propositional forms of Brentano to build his diagrammatic system, rather than the traditional Aristotelian forms, i.e. all universal propositions are phrased in terms of what they deny, rather than what they affirm.
And, to echo Arthur Prior’s point above, Venn states that this interpretation …enables us to express each separate accretion of knowledge, and so to break up any complicated group of data, and attack them in detail.
^{[14]}
Euler vs. Venn Diagrams
In 1880, John Venn, the English mathematician and logician, published a paper On the Diagrammatic and Mechanical Representation of Propositions and Reasonings, introducing his revolutionary logical diagrams.
While his approach was completely novel, he was continuing a long tradition of logic diagrams in Europe.
Leonard Euler, the famous mathematician, did much to popularize such graphical aids in his Lettres à une Princess d’Allemagne.
This was a compilation of letters he sent (on various topics) to Friederike Charlotte of BrandenburgSchwedt and her younger sister Louise, between 1760 and 1762.
The diagrams he used to explain syllogisms became known as “Euler diagrams”, and with the widespread popularity of his letters to the young princesses, they become the default standard in Europe.
Euler’s 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.
So, given the traditional proposition, A: “All Greeks are Europeans” the corresponding Euler diagram would be:
Euler’s diagrams were founded on the traditional logic, and the main aspects are:
 Euler’s system represents propositions, with each circle (or part circle) directly representing a term of the proposition.
 His propositions have the traditional form.
 Eulerian circles are only drawn if the terms of the proposition exist, and they may or may not overlap. If the circles overlap an “*” is used to indicate there is at least one member of the class in that subpart.
As discussed, modern logic is founded on very different grounds to traditional logic and takes the BrentanoVenn interpretation of propositions.
For example, modern logic has the universal positive proposition, A_{m}: “There does not exist a Greek that is not European” which we can represent using the following Venn diagram:
Since Venn’s diagrams are founded on modern logic, they differ in key aspects from Euler’s diagrams:
 Venn’s system represents classes of things directly, rather than the Eulerian approach of representing propositions (which then relate classes of things).
 He uses the BrentanoVenn form of propositions.
 Vennian circles always overlap with each other, and their subparts are modified by:
 shading if they are empty,
 marking with an “×” to indicate there is at least one member of the class in that subpart.
I’ll repeat: in a Venn diagram, we shade an area not to draw attention to it, but to indicate that the compartment (or type of thing) does not exist.
The differences between the two diagrammatic systems should be no surprise—merely reflecting the fundamental differences between the traditional and modern logics.
Modern Square of Opposition
As discussed, the Square of Opposition is a visual depiction of the inferential relationship between the four fundamental propositions of Aristotle (traditionally labelled as A, I, E and O).
Modern logic has its own version of the Square of Opposition.
But it looks a little sparse compared to the traditional one.
Why?
Well, because the alternative propositions of modern logic (A_{m}, I_{m}, E_{m} and O_{m}) do not have the same truthrelations between them as their traditional forms do.
If you change the logical rules, expect different logical results.
And what are these differences you ask?
Well, let me take it from the top.
The top of the Square of Opposition, that is.
Namely, the contrary relation between the universals of modern logic.
The contrary relation between the universal propositions (A_{m} and E_{m}) can be summed up as: “not both are true”.
If one is true, the other must be false–and both may be false.
Recall, that for A_{m} we have: “There does not exist an S that is not P”
And for E_{m}, we have: “There does not exist an S that is P”
Now, if S does not exist, then both A_{m} and E_{m} are true.
And rightfully so, since both their claims are true, given that no S exists to contradict either of them.
Thus, the claim of the contrary relation that “not both are true”, no longer applies.
Both can be true in modern logic.
So byebye contrary!
Next, the subcontrary relation, the one that lies under (Latin: sub) the contrary relation in the Square of Opposition.
The subcontrary relation between propositions I and O can be summed up as: “at least one must be true”.
Both can be true, but at least one must be true.
Recall, that for I_{m} we have: “There exists an S that is P”
And for O_{m}, we have: “There exists an S that is not P”
Now, if S does not exist, then both I and O are false.
And rightfully so, since there is no S that we can predicate anything of: there is no S that is P, nor any S that is not P.
Thus, the claim of the subcontrary relation that “at least one must be true”, no longer applies.
Byebye subcontrary!
Next, we’ll remove the subaltern relation, which exists between the universal propositions (A_{m} and E_{m}) and their respective particulars (I_{m} and O_{m}).
The subaltern relation between universal and particulars can be summed up as: “if the universal is true, then the particular is true”.
This is quite intuitive: what is true for the whole of a thing, is true for any particular part of the whole.
In Aristotle’s logic, this claim is known as the dictum de omni et nullo.
(Latin for “the maxim of all and none”)
I’ll focus on the positive propositions A_{m} and I_{m}, but my analysis applies equally to the negative propositions, E_{m} and O_{m}.
Recall, that for A_{m} we have: “There does not exist an S that is not P”
And for I_{m}, we have: “There exists an S that is P”
Now, if S does not exist, then A_{m} is true, but I_{m} is false.
And rightfully so, since A_{m} makes a negative claim about the existence of S (S does not exist), while I_{m} makes a positive claim (that S does exist).
Thus, the subaltern claim that “if the universal is true, then the particular is true” no longer applies.
Byebye subaltern!
Finally, we’ll remove the superaltern relation, which exists between the particular propositions (I_{m} and O_{m}) and their respective universals (A_{m} and E_{m}).
The superaltern relation between particulars and universals can be summed up as: “if the particular is false, then the universal is false”.
This is quite intuitive: if there is not one single part of a thing that has a given property, then the thing (as a whole) lacks that property.
I’ll focus on the positive propositions I_{m} and A_{m}, but my analysis applies equally to the negative propositions, E_{m} and O_{m}.
Recall, that for I_{m} we have: “There exists an S that is P”
And for A_{m}, we have: “There does not exist an S that is not P”
Now, if S does not exist, then I_{m} is false, but A_{m} is still true.
And rightfully so, since I_{m} makes a positive claim about the existence of S (that S exists); A_{m} makes a negative claim about the existence of S, and remains true (since no S exists that is not P).
Thus, the superaltern claim that “if the particular is false, then universal is false” no longer applies.
Byebye superaltern!
And so we are left with the only remaining relation between propositions: the contradictory relation between A_{m} and O_{m}, and between E_{m} and I_{m}.
Summed up, contradiction means “one or the other, but not both”.
I’ll stick with A_{m} and O_{m}, but my analysis applies equally to the other contradictory pair E_{m} and I_{m}.
Recall, that for A_{m} we have: “There does not exist an S that is not P”
And for O_{m}: “There exists an S that is not P”
You can see that both propositions cannot be true at the same time.
Only one of them can be true—and one of them has to be true.
For any S, if it exists, it either:
 has the property P (and A_{m} is true), or
 it does not have the property P (and O_{m} is true).
And if S does not exist, then A_{m} is true and O_{m} is false.
Thus, the claim of the contradictory relation that “one or the other, but not both” always applies.
So, hello contradictory!
In modern logic, the only inferential relation between propositions that still applies, is the contradictory relation of “one or the other, but not both”.
So, the two systems of logic (traditional and modern) are completely incompatible in regard to the inferential relation of their respective propositions.
This is fine.
Different logical rules, different logical results.∎
Footnotes

The “Barbarous” Barbara Celerant.
Each artificial word (in italics) of this nonsense rhyme contains three vowels, which indicate the type of propositions (A, E, I and O), and represents a valid syllogistic argument.
Each syllogism has a particular arrangement of its three terms (minor, middle and major), and the arrangement determines if the syllogism is valid, i.e. it is correct in its inference.
There are four possible arrangements of terms in syllogisms, called “figures”, and the type of propositions involved set the “mood” of the syllogism.
Thus, “Barbara” indicates a valid mood of the first figure, which has:
 A for the first premise,
 A for the second, and
 A for the conclusion.
(We might write “Barbara” as bArbArA to make this relation clearer)
An example of a first figure syllogism, in the mood of Barbara: A: “All Mammals are Vertebrates” A: “All Dogs are Mammals” A: “All Dogs are Vertebrates”
As we can see, this is a valid inference.
The rhyme lists all nineteen valid arguments of the traditional syllogism and contains letters indicating how the different figures can be converted between one another.
Allinall, it is a very useful mnemonic tool.
Key Takeaways
 Logic is about reasoning: the process of going from what we know (the premises) to some new knowledge (the conclusion), via indirect means.
 An inference is an individual step in the reasoning process, and inferential systems are formal systems studied for their logical properties.
 Aristotelian logic refers to the body of logic developed by Aristotle, during the 4^{th} century BC. He was the first to study abstract patterns of logical inference and invented the syllogisms: arrangements of terms that guarantee correct conclusions.
 Traditional logic was developed by the Medieval Scholastics: they catalogued, analyzed and developed the work of Aristotle.
 Modern logic was created from the 1840s onwards. Heavily influenced by mathematical symbolism and mechanical processing, its developers shared a desire to make logic a completely formal science.
 Squares of opposition are used to show the immediate inferential relations between propositions, e.g. if one is true, can another be true as well?
 Euler diagrams, based on traditional logic, show only the actual logical relations between existing things.
 Venn diagrams, based on modern logic, show all possible relations between things—and then shadeout those things or relations that do not exist.
References
 Londey, D. & Johanson, C. 1984. Apuleius and the Square of Opposition. Phronesis, Vol. 29, No. 2, pp. 165–173. [Back]
 De Rijk, L. M. 1964. On the chronology of Boethius’ works on logic II. Vivarium, Vol. 2, Issue 1, pp. 125–161. [Back]
 King, P. 2007. Boethius: The First of the Scholastics. Carmina Philosophiae, Vol. 16, Part 1, pp. 23–50. Available at: http://www.individual.utoronto.ca/….
(Accessed: 19^{th} July 2024). [Back]  Vandoulakis, I. M., & Denisova, T. Y. 2020. On the Historical Transformations of the Square of Opposition as Semiotic Object. Logica Universalis, Vol. 14, pp. 7–26. [Back]
 Boethius c. 505. De Syllogismo Categorico. Edited by Migne, J. P. 1847. Patrologia Latina, Vol. 64, p. 799C. Available at: https://documentacatholicaomnia.eu/… (Accessed: 20^{th} July 2024). [Back]
 Aristotle. Peri Hermeneias. Translated by: Boethius c. 515. De Interpretatione: De subjectis et praedicatis enuntiationum. Edited by Migne, J. P. 1847. Patrologia Latina, Vol. 64, pp 323D–324A.
Available at: https://documentacatholicaomnia.eu/…
(Accessed: 20^{th} July 2024). [Back] 
Jevons, W. S. 1888. Elementary Lessons in Logic: Deductive and Inductive, pp. 144–145. London and New York: MacMillan and Co.
Available at: https://cdn.mises.org/…
(Accessed: 20^{th} July 2024). [Back] Ibid., p. 145. [Back]
 Jevons, W. S. 1869. The Substitution of Similars: the True Principle of Reasoning, Derived from a Modification of Aristotle’s Dictum, p. 43. London: MacMillan and Co. [Back]
 Brentano, F. 1874. Psychologie vom empirischen Standpunkt. Leipzig: Duncker & Humblot. [Back]
 Prior, A. N. 1976. The Doctrine of Propositions and Terms, p. 123. London: Duckworth. [Back]
 Land, J. P. N. 1876. Brentano’s Logical Innovations. Mind, Vol. 1 (original series), Issue 2, p. 292. [Back]
 Joyce, G. H. 1916. Principles Of Logic, p. 117. London: Longmans, Green & Co. [Back]
 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/…
(Accessed: 20^{th} July 2024). [Back]  Ibid., p. 11. [Back]