A More General Theory of the Syllogism

in which it is shown that
Aristotle missed half of them,
there's nothing to them, and
we can do better without.


This essay on the unnecessary complexity of Aristotelian syllogistic logic is part of my attempt to reduce unnecessary intellectual complexity in science, philosophy, and emotional self-management. In general, we could make things a lot less complicated for ourselves.

Why so complicated? On the one hand, people sometimes only figure things out at the beginning in a more complex kind of way, through some initial inkling from some partial perspective, and maybe they don't see the simpler insight until later, or not at all. On the other hand, complexity serves the interests of a professional class, as it supports their job security by keeping out competitors, and holding down their customers, who could otherwise figure out for themselves how to do that professional service, including lawyers, plumbers, doctors, and as it turns out, medieval logicians.

Aristotle, the greatest thinker in history, was considered, more or less, the beginning and end of all knowledge for thousands of years.

I feel an emotional affinity with Aristotle through his very name, which is aristo "best", + tele "purpose". The purpose of my own life is, has always been, exactly that (unknown) best purpose. Trying one's best over a lifetime, to more deeply understand, and then to share understanding with others, is that best purpose, it seems. We both found the best education there was in our time, and we both somehow have permission to not specialize, but to think and write about an unlimited variety of topics, we both gave most of our lives to it. How satisfying, true, and beautiful this is, for me.

Aristotle wrote on subjects I too find interesting, including biology, physics, metaphysics, poetry, ethics, psychology, linguistics, economics, and logic (links are to some of my efforts in those areas; I celebrate sharing so many interests), but his works are comprehensive, and definitive, for scholars up until the Renaissance.

Now, much of Aristotle's logic was lost to Europe before Averroes/Ibn Rushd "The Commentator" made a lot of detailed explanations for his Muslim king in Spain, the one who said This Aristotle guy isn't very comprehensible, is he? Then Averroes gave short, medium, and long explanations for all of Aristotle's works that he had, which were a bunch more than they had in Europe, and those got translated into Latin, so, as it turns out, the founding ideas of Renaissance Europe weren't even in Europe during the entire middle ages and had to get rescued by the Arabs. Europeans and Westerners should be humble and grateful, contemplating the unity of history, instead of resentful or thinking they are all superior, contemplating only an imagined separate uniqueness, as for many generations we have done.
His work on logic, The Organon, essentially created the entire field of logic; he took it to such a high state that it was hardly improved for thousands of years, even Kant in the 1700's said no-one had improved Aristotle's logic before his own time; that would be, indeed, about 2000 years. Thus although there were previous general questioners in philosophy, like Parmenides using the "reductio ad absurdum" method, or Plato asking what is the nature of true and false, what is the connection between the premises and conclusions in an argument, and what is the nature of definition, it was Aristotle who really made it a thing. He created a rich and comprehensive formal system capable of expressing any logical relationship and deriving a valid and reliable conclusions from them.

The foundation of Aristotle's logic are his four connectives and the syllogism. It is an enjoyable distraction for a modern philosopher to learn Aristotle's connectives, to give them a meaning in modern understanding, and to reason through his syllogisms, patterns of deduction from premises to conclusions. I will offer that for you, here, and incidentally show how he seems to have missed a dozen valid syllogism patterns (perhaps due to theorizing too early, getting in the way of himself with arbitrary and unnecessary restrictions); also that two of his syllogisms are invalid in principle. I began this as a schoolboy exercise to simply understand Aristotle's logic, and was surprised to see all what fell out. Perhaps another scholar wrote or knew these findings; I plead ignorance. But since I haven't found by others what I found by myself, here we go.

Aristotle's connectives are quite similar to modern logical connectives like "and", "or", "implies". They are four, and, simply enough, they assert overlap and non-overlap, inclusion, and (partial) exclusion between concepts. He wanted some specifically logical language to express them in, so he invented one, like Frege invented ∀ and ∃, and like Panini invented his rule formalism for Sanskrit around that time, but Aristotle was less adventurous in symbolism so he picked four vowels, to be the names of these relationships. Conveniently he picked a subset of his ancient Greek alphabet that remains accessible today, four of its vowels, alpha, epsilon, iota, and omicron, which were imported without change into the Roman alphabet as a, e, i, and o, and handed down to us. How wise and friendly he was, to use basic sounds to indicate basic concepts.

LetterLatin mnemonicMy mnemonicExampleDefinition by AristotleDefinition 2
aaffirmoallAaBA belongs to all BEvery B is an A
enegoemptyAeBA belongs to no BNo B is an A
iaffirmo(partly) inAiBA belongs to some BSome B is an A
onego(partly) outAoBA does not belong to some BSome B is not an A

A and B are arbitrary terms, ideas, concepts, categories, etc., which are conceptual things of any character except that they let you say yes or no. "Term" is an extremely general and generic class of ideas. For example across a universe of discourse in particular cases, you could say Yes this example is in this category or this term applies, or No it doesn't. Or they could be bits of information, a 1 or 0 in a particular digital sensor, or a classifying mechanism, or a 1-0-1-0 alternating ultra-simple clock, or a random bit generator, or whatever you like, it just needs to produce yes and no somehow or other.

Let's see how the Letters relate to the logical connectives used nowadays. Nowadays we have two symbolisms for exhaustive, therefore correct, reasoning about relationships between ideas or possibilities, one as a graph of one or more bounded regions, another as a list, that is, as Venn Diagram and as truth table. They correspond exactly.

I will refer to it here as a possibilities table, since here we aren't putting truth values into it but possibilities of yes or no as to whether some category applies to some thing or things or not. We will want to rule in or rule out possibilities which may involve certain combinations of Yeses or Noes, and such assertions would confuse me if the No meant False but the ruling out also meant False. So possibilities table it is. But it is simply a table of all binary combinations, and the main point of it is that it exhausts all the possibilities.

 < == >
 A    B 
 y    y
 y    n
 n    y
 n    n

All four logical possibilities are visible in, and the point of, both the Venn diagram and the possibility table. For each region of the Venn diagram, either A covers it, applies to it, or not, and either B covers it or not. Similarly, every row in the possibility table has a Yes or a No for A, and a Yes or a No for B. Those are all possible combinations.

Footnote: You could deny this by denying the law of the excluded middle, by saying something can in logic be neither A nor not A. But that just gives three options for each, so there would be 9 rather than 4 rows in the possibility table, and perhaps the Venn diagram would look a bit more like a grid. And all these rules would need a review to see if they still hold. Another option, going even farther, is to use Fuzzy Logic and assert that set membership can be of any fractional degree between 0 and 1 inclusive; then a Venn diagram would have to be a two dimensional grayscale, and a possibility table would seem to be some set of equations rather than a list of discrete rows. But one does like to keep things simple, so for now, let's.

The graphical form lets us see the global relationships instantly and intuitively, while the table form lets us work through the possibilities carefully, individually, and exhaustively, so as to be sure we have missed nothing, to be sure that we are right. These tools give us a lot of intellectual power, because we can consider all the possible combinations when we think about some question, and if we have an answer for each one, then we have a total and 100% answer for our question, because we have an answer for every possibility. That is the infinite and utter power of tautology. (H!)

Now Aristotle's Letters, these logical symbols or terms, can be very easily, and equivalently (without loss or gain of significant information) defined on both the possibility table and the Venn diagram.

Each Letter represents a concrete assertion, that there is definitely nothing, or definitely something, at one place or another in the space of possibilities, whether drawn as a diagram or listed in a table. The next few pictures and paragraphs, the rest of this section will just repeat the last sentence in detail.

So the letters 'a' and 'e' assert that nothing satisfies a certain row in the table of possibilities.

AaB means there is nothing that is B that is Not A.
AeB means there is nothing that is both B and A.

Letters 'i' and 'o' assert that something does satisfy a certain row in the possibility table:

AiB means there is something that is both A and B.
AoB means there is something that is B that is not A.

I found I could express all of these as annotations on a single table, as follows:

 A    B 
AeB X y    y+ AiB
 y    n
AaB: X n    y+ AoB
 n    n

"X" means "there exists nothing in the universe of discourse satisfying this row."
"+" means "there exists something in the universe of discourse satisfying this row."

A vision of logical power, indeed, some excitement, comes from the above table. 'e' and 'i' are related (as opposites), because they are based on the same row in the table. In the same way, 'a' and 'o' are related as opposites, based on their same row. Then, 'a' and 'e' are the same, both using the "nothing matches this row" assertion, but with different rows, and 'i' and 'o' are also the same, both using the "something matches this row" assertion, but again with different rows. I think this view is rather more accessible than the medieval Square of Opposition, which contains the same information but without being so directly understandable, since you have to go back to first principles to verify each statement there, whereas here, defining the letters on the table of possibilities means that you can see the relationships directly.

Observe next that the rows chosen for these letters include a row with sames, and a row with differents. Row one has Y and Y; row three has N and Y. It is so very elegant, since not just those two rows but all the rows have either sames or differents. With AND, OR, NOT, and parentheses for order of evaluation, added to the logical armamentarium, one has a complete language for asserting any combination of any of the rows in the table of possibilities. We will return to this below.

For now, though, let's see how these assertions about table rows relate to Venn diagrams, one for each of the four letters 'a', 'e', 'i', and 'o'.

First, in the case of AaB, the B-outside-A zone does not exist: there is nothing in it, because B is entirely inside A, as drawn. Similarly in the possibility table, we have an X in the row where A is a No and B is a Yes, meaning that nothing can be both Not-A and B.

 < == >
 A    B 
 y    y
 y    n
AaB: X n    y
 n    n

See how the table and the diagram express the same information?

Second, in the case of AeB, the B-inside-A zone does not exist: there is nothing in it because B doesn't overlap with A.

 < == >
 A    B 
AeB X y    y
 y    n
 n    y
 n    n

Third, in AiB, the B-inside-A zone is populated, there is at least something existing in there. In set theoretic terms A∩B≠∅.

 < == >
 A    B 
 y    y+ AiB
 y    n
 n    y
 n    n

Fourth and finally, in AoB, the B-outside-A zone is populated, there is at least something existing in B and not in A: ¬A ∩ B ≠ ∅.

 < == >
 A    B 
 y    y
 y    n
 n    y+ AoB
 n    n

Now that we have Aristotle's logical letters defined on the table of possibilities, we can ask, what are these possibilities, and what if there is a third? What indeed are the possibilities A and B we are using here? Aristotle likes to prematurely restrict his theory by saying A must be a Subject and B must be a Predicate. However the same logical relationships and provable outcomes follow identically whether these placeholder names, A and B, hold the place for Subjects, Predicates, Categories, Sets-by-listed-membership, or Sets-by-membership-function, etc. Aristotle and after him everyone that followed him for 2000 years all get tied up into knots trying to figure out how to extend and tighten up the description of all this symmetry by saying well let's say it only applies to Subject-Predicate assertions of a particular type or in a particular order, and such, thereby missing the point of a truly general system. So the possibilities might be categories that things can be inside of or outside of, or they might be subjects which might be sets of things, or predicates which need a subject to apply to and then applies or doesn't apply to it, or things that exist or don't exist. Aristotle got really wrapped around the axle by existence versus non-existence. As we will see.

My point is that less is more. We don't have to specify all these things that just make our minds more confused, when the simpler, more general, more obvious, and more applicable essence is ready to hand and takes us farther and faster than we can go when we are overburdened with all these irrelevancies that just produce paradoxes and difficulty. Aristotle, dude, let Frege handle Subject and Predicate, okay?

The Syllogism

Now a "syllogism" is a general rule of deduction in which, given two (letter-based) premises, one conclusion may be drawn. Aristotle restricted the definition unnecessarily to only two premises that mention the two terms of the conclusion, each once, and one other term (called the middle term), twice. And he further economized too early by sorting them in order of the term labels, into "First, Second, Third, and Fourth Figures", which lets us figure out what term occurs before or after the logical letter in first or second premise or in the conclusion.

First Figure means AxB, BxC ⇒ AxC
Second Figure means AxB, AxC ⇒ BxC
Third Figure means AxC, BxC ⇒ AxB
Fourth Figure means BxA, CxB ⇒ AxC

So AaB, BaC ⇒ AaC is a First Figure Syllogism. They gave it the name "Barbara". In fact, they were mnemonic crazy, they gave names to all the syllogisms they came up with. These are Latin-like words, where the first, second, and third vowels in the name are the letters in the first and second premise and (third) in the conclusion. So if you know the name and the figure, you can figure out the whole thing. Memorizeable, yes. Simple: no. Complete: no. Always valid: no.

So before we go into all the rules, let's just calm down a moment and realize these are just people like us, their minds are no smarter nor intrinsically capable of getting deeper insights than ours are. In fact, the reason this entire subject makes any sense at all, or has the least bit of validity, is that the reasoning itself is ultimately so simple and perfect and impossible to deny if you just look at even a simple picture like a Venn diagram or a truth table with just a bit of understanding, then if you can go back to the definitions (which in this case are very useful definitions), then everything falls out and makes perfect sense and you can walk the tightrope of truth without any errors or failures or falls.

I'm telling you this because I'm about to take you through the tautological (obvious, undeniable, too simple for words) reasoning process that proves not only all of Aristotle's syllogisms (the ones that are true), (and shows how the ones that aren't, aren't), but also adds another dozen valid syllogisms to the list. Because simpler is better, and if we can make things simpler, we might be able to go farther, faster, with more confidence than when we have to struggle to take each step of understanding or communication.

We just have to break it down to the simplest possible things.

So we have these annotations on the possibility table, right? The first thing to do is, consider what happens to that table, and what the letters will look like, when we have three categories or terms. Instead of just two, with placeholder names A and B, what are all the possibilities if there are three? Four would be another question, and we won't get into that but it's certainly imaginable.

Let's use C as an arbitrary placeholder label for the third category or "term". We don't care if it's a subject or predicate, a term or a category, it's arbitrary and all we will ask of it is for it to have yes and/or no associated with it somehow. Which amounts to no demand at all.

We will build the 3 category possibility table in the same way that we build a 2-category table. We start with nothing, a table of no columns and no rows, representing everything by saying nothing. Then we pick up A, and we figure that A (or "being A" or "having A" etc.) can be a yes or a no. That's all the possibilities for A, right? Any questions there? In fact, did I even say anything, really, there? No, I would say I didn't actually say anything. Because Is and Isn't are the logical possibilities, is the definition of logical possibility. It can't be wrong, because it's just a definition. It could fail to be useful, but it can't be wrong. And it doesn't say anything, to make up a definition and assert that that's a definition. People might differ, and some definitions could be useless, but a useful definition isn't more falsifiable because it is useful. Definitions are non-arguable. And incidentally this one is quite useful, so we are going with it.

Okay, so in the one category table with just A, there are two rows, a yes row and a no row. And we continue. For each of the possibilities of A, there are two possibilities for B, namely a yes or a no for B. We can draw them with loops on the Venn diagram or make a table in some sorted order, or whatever we like, that's all the possibilities there are. So that's our 2-category table, with four rows YY, YN, NY, and NN, and there is no possibility outside the list, no matter what A and B are (that is, as long as the excluded middle applies).

We can repeat this process. For each of the combined possibilities of the categories enumerated so far, there are exactly two possibilities for the next category, namely, a yes or a no, in this case for category C. And that's all that there are. The possibility table can be written like this, and I will number the rows for convenience so we can refer to them later.

Row A    B   C 
1 y    y  y
2 y    y  n
3 y    n  y
4 y    n  n
5 n    y  y
6 n    y  n
7 n    n  y
8 n    n  n

So this is our tidy table of all combinations of A, B, and C. You could sort the rows differently, it wouldn't matter, the row numbers would be different but the ideas would be the same. Of course. (Have some confidence: every step should make so much sense that you can't see it otherwise.)

We can also represent all the same possibilities graphically, as in this Venn diagram.

Each number in the above figure identifies one row of the possibilities table with the same Yes and No pattern. Yes for In a labelled region, No for Out of the labelled region. A region in the diagram and a row in the table represent one and the same logical pattern. We know this already, but this just says what we can do with two categories, we can also do with three.

Our next task, following Aristotle now, will be to represent all the connectives 'a', 'e', 'i', 'o', with all the terms A, B, and C (in that order), showing exactly what they are saying about the table of possibilities. Again X means that row is empty: nothing can exist there. And + means those rows together are not empty: something exists in at least one of them (otherwise, then the original definition of the letter could not apply.) I'll give you a whole table for each possible assertion, which is 4 x 3 tables. They will be convenient later when we combine different assertions and see if any other ones fall out automatically, (and they do).

#AaB
y y y
y y n
y n y
y n n
n y y X AaB
n y n X AaB
n n y
n n n

# A    B   C AeB
1 y y y X AeB
2 y y n X AeB
3 y n y
4 y n n
5 n y y
6 n y n
7 n n y
8 n n n

# A    B   C AiB
1 y y y + AiB
2 y y n + AiB
3 y n y
4 y n n
5 n y y
6 n y n
7 n n y
8 n n n

# A    B   C AoB
1 y y y
2 y y n
3 y n y
4 y n n
5 n y y + AoB
6 n y n + AoB
7 n n y
8 n n n

# A    B   C AaC
1 y y y
2 y y n
3 y n y
4 y n n
5 n y y X AaC
6 n y n
7 n n y X AaC
8 n n n

# A    B   C AeC
1 y y y X AeC
2 y y n
3 y n y X AeC
4 y n n
5 n y y
6 n y n
7 n n y
8 n n n

# A    B   C AiC
1 y y y + AiC
2 y y n
3 y n y + AiC
4 y n n
5 n y y
6 n y n
7 n n y
8 n n n

# A    B   C AoC
1 y y y
2 y y n
3 y n y
4 y n n
5 n y y + AoC
6 n y n
7 n n y + AoC
8 n n n

# A    B   C BaC
1 y y y
2 y y n
3 y n y X BaC
4 y n n
5 n y y
6 n y n
7 n n y X BaC
8 n n n

# A    B   C BeC
1 y y y X BeC
2 y y n
3 y n y
4 y n n
5 n y y X BeC
6 n y n
7 n n y
8 n n n

# A    B   C BiC
1 y y y + BiC
2 y y n
3 y n y
4 y n n
5 n y y + BiC
6 n y n
7 n n y
8 n n n

# A    B   C BoC
1 y y y
2 y y n
3 y n y + BoC
4 y n n
5 n y y
6 n y n
7 n n y + BoC
8 n n n

Okay now things are getting hot now. We are about to prove all of Aristotle's syllogisms, in short order. There are two ways we can do it, the "Presto" or double cross out method, and the row rule method.

In the double cross out method, look around in the above tables for any of the 'a' or 'e' letters. Those cross out two rows in the table, do you see? For example, AaB crosses out rows 5 and 6, it asserts that whatever the value of C might be, there is nothing that is B which is not also A. I could make up ten ways to say the same thing and it would only get more complicated. The simple thing is, AaB crosses out rows 5 and 6, all the rows where A is no, and B is yes.

Next step. Pick another cross-out example. Let's use BaC. That crosses out rows 3 and 7 from the possibilities table; as you recall, BaC means All C's are B's, or There is no C which is not B, or, well, it means BaC.

Suppose we combine these two, then, and make both of two assertions, both of AaC and BaC, and the information we get from that is that rows 5 and 6 are crossed out by AaC, and rows 3 and 7 are crossed out by BaC. In all, 3, 5, 6, and 7 are crossed out: none of those particular combinations can apply if AaC and BaC are given as true. Right? It looks complicated but it's not. Do you see?

Next step: Did we discover something? Well, yes, if you look around all those tables, you might discover that another assertion, specifically AaC, says that 5 and 7 are crossed out. But don't we know that if 3, 5, 6, and 7 are crossed out, then 5 and 7 crossed out? Well, duh, yes they are. H. So doesn't that mean exactly that if AaB and BaC are given, then AaC follows? I mean, is there any other option? No, I don't think so. You can go through the reasoning on a possibilities table, or a (complicated) Venn diagram, or in heavy cogitation on verbal sentences expressing some subjects and predicates exemplifying these premises and conclusions, and they will all track exactly parallel, because each represents the others in logic.

Convince yourself. You will discover calm.

So that's how we roll. Pick out any two X-out tables above, and combine the four X'ed out rows from them, and see if those occur in any of the other X-out tables. For lack of a better phrase, I'm calling it the double cross-out method. If you find three cross-out tables, such that if you combine two of them, they cross out both of the rows that are crossed out in the third, then the assertions represented by the conclusions are already part of the assertions in the premises, and therefore the conclusion follows from the premises. You can't have the premises without having the conclusion.

I have done this counting on my thumbs, and the ones Aristotle also found are these:

AaB (5X,6X), BaC (3X,7X) ⇒ AaC (5X, 7X) ("Barbara")
AeB (1X,2X), BaC (3X,7X) ⇒ AeC (1X, 3X) ("Celarent")
AaB (5X,6X), AeC (1X,3X) ⇒ BeC (5X, 7X) ("Camestres")
AeB (1X,2X), AaC (5X,7X) ⇒ BeC (5X, 7X) ("Cesare")

And here are some more, which don't apparently have names. I seem to have discovered four new syllogisms unknown to Aristotle.

BaC (3X,7X), BeC (1X,5X) ⇒ AaC (5X, 7X)
BaC (3X,7X), BeC (1X,5X) ⇒ AeC (1X, 3X)

AaC (5X,7X), AeC (1X,3X) ⇒ BaC (3X, 7X)
AaC (5X,7X), AeC (1X,3X) ⇒ BeC (1X, 5X)

AaB (5X,6X), AeB (1X,2X) ⇒ BeC (1X, 5X) (and CaB, but we'll ignore that as being out of order)

Well the reason, of course, that these syllogisms may be unknown to Aristotle, and perhaps to history itself, is because they don't fit into his tidy (I would say unnecessarily tidy) list of Figures, with A and B and C in the right places.

Here we have a term in the conclusion that wasn't even mentioned in the premises, how could that be? Aristotle might complain, how can you infer something about the relationship between B and C for example, when neither of the premises even give a mention of one of them?

Here's how. All these take the form of having both 'a' and 'e' between the same two categories. Which is to say, in the 'a' case, positively, one contains the other, and in the 'e' case, there is nothing in the intersection. The 'a' assertion says nothing in the second exists outside the first, while the 'e' assertion says nothing in the second exists inside the first. This seems like a bit of a conundrum until you realize that the second category may be entirely empty, like "Witches" or "Magic". Here would be an example, then:

Let A be magic, and B be things in my house, and C be things in your house. Then the argument, AaB, AeB ⇒ AeC, can be translated into words like this: We are given the proposition AaB that there is no magic outside my house, and AeB that there is no magic inside my house. Then the proposition follows, AeC, that there is no magic in your house.

This makes sense because there no actual magic anywhere, therefore it's not inside my house nor outside of it (and those being all the possibilities, relative to my house anyway), so it can't be in your house either. (You can have magic in your imaginary world if you want, but then in that world you have to deny AaB or AeB.) See how we created a proposition about your house, about something we know nothing about, and said nothing about, but then proved it to be true?

That would seem to be how these additional four syllogism patterns or "moods" got passed over. They aren't in the tidy lists of Figures with A's, B's, and C's in the right number in the right order in the right places. But that doesn't make them any less perfect, true, eternal, valid, and general. Nor does another name other than syllogism come reasonably to mind, considering that syllogisms and these, both, combine two premises and a conclusion, each based on the four letters 'a', 'e', 'i', 'o', each using three arbitrary categories or terms, and each deriving a logically necessary conclusion from those two premises.

I suppose Aristotle might counterargue by saying that these new ones don't count because the empty set is not a set, or doesn't exist, or something like that, because every category has to have things in it. That seems to have been the stumbling block for ancient logicians. Even imaginary categories, like magic? I don't see why we can't reason about imaginary things. I'm a logic imperialist. Logic rules everywhere, from before the beginning to after the end, so why not also in imagination land, to the extent that is consistent.

Now I would still criticize the four new double-cross syllogisms, myself, on the grounds of equivalence. If you relabel A as B or B as C, etc., you can turn some of these into others, and so they are really fewer rules than four, maybe even just one. I leave this to the reader as an exercise.

So now we are done with the double cross method, and it's time to turn to the "row rules" method. Thanks for your patience! Maybe take a break, have a cup of tea and a cookie, and when logic inspires you again, come back.

Now then, the row rules method of syllogism construction, and proof, goes like this. Pick one of the rows of the 8-row possibilities table. Any one will do. (Well, not quite; the even numbered rows seem to be unproductive.)

Recall now that the assertions in 'i' and 'o' say that there exists something that satisfies certain row(s). AiB for example says that there is something in row 1 or row 2, because irrespective of C, there is something that is both A and B; row 1 and 2 differ only in whether they satisfy C or not, so both are places where AiB could be satisfied: At least one thing is in there, there could be more, and it could fit in row 1, or in row 2, or there could be some in both, but rows 1 and 2 together will not be both completely empty with nothing in either of them. That's a lot of words for AiB, increasing the noise around the concept, so please refer back to the definitional table and think about it if you need to.

So this means that if one of the rows turns out to be empty, then the other one has to have something in it. If you say 'i' or 'o', which asserts something in (at least) one of the (multiple) rows that it has tagged, and you come along later and somehow show that one of those rows is empty, then you may properly conclude that the remaining row is not empty. This is also called the process of elimination, which is a rule of logic that noone ever has to explain, teach, or even learn. It's tautological. H. If the cat is not in any room in the house, then by process of elimination, it must be outside the house. Again, does the process of elimination say anything? I assert that it doesn't. (It's tautological. H.)

So let's apply this reasoning to the rows of the possibilities table. We have six different assertions, AiB AoB BiC Boc AiC AoC, each stating that some pair of rows is not empty, there is something there in at least one of them. Great.

Pick one, any one, and show it to me. So its pair of rows are say R1 and R2. This will not be magic, I promise.

Next, look at all the cross-out tables with the 'a' and 'e' assertions, and find one that crosses out R1 or R2 but not both of them. Let's say, without loss of generality that it crosses out R2. Then by process of elimination, R1 has something in it. Great.

So here's the trick. If we know something is in one row, and some assertion happens to say that there must be something in that row or another row, then it would be a true and correct assertion, because being in that row it is certainly in that row OR another row. We sort of lose information, by going to that latter assertion. It's not as specific as the knowledge that we already had, because it doesn't specify which row is populated, even though we happen to know the answer to that question. We don't have to tell anyone, necessarily, but the latter assertion can certainly be held to be true, based on our premises.

That's it. Two rows having something in one by an 'i' or 'o' statement, one of them having nothing in it according to an 'a' or 'e' statement, therefore the remaining row by process of elimination must have something in it, and another 'i' or 'o' statement which includes the row in question is fully satisfied, even with the loss of information attendant to only saying the conclusion, still the conclusion is true and undeniable. That's our row rules method, which relies on the equivalence of the possibilities in the possibilities table and the possibilities that are possible, to show that the combinations labelled with the Aristotelian letters have the relationships which the syllogisms declare. (That's a fancy way of saying nothing, by the way, as in "Two names for the same thing each name the same thing." It might make you feel better sometimes to double the word count and snow the audience, but the inner insight is the simple part that you can really hang on to if you want to have confidence in your own idea. So hold on to the inner insight.)

Here are some syllogisms proved by the row rules method which Aristotle gives us:

AaB (5X,6X), BiC (1∨5) ⇒ AiC (1∨3) ("Darii") (Row 1)
AaC (5X,7X), BiC (1∨5) ⇒ AiB (1∨2) ("Datisi") (Row 1)
AiC (1∨3), BaC (3X,7X) ⇒ AiB (1∨2) ("Disamis") (Row 1)
AiC (1∨3), AeB (1X, 2X) ⇒ BoC (3∨7) ("Festino") (Row 3)
AoC (5∨7), BaC (3X,7X) ⇒ AoB (5∨6) ("Bocardo") (Row 5)
AeC (1X,3X), BiC (1∨5) ⇒ AoB (5∨6) ("Ferison") (Row 5)
AeB (1X, 2X), BiC (1∨5) ⇒ AoB (5∨6) ("Ferio") (Row 5)
AaB (5X,6X), AoC (5∨7) ⇒ BoC (3∨7) ("Baroco") (Row 7)

And here are some which I have not found in Aristotle. I'm no Aristotle scholar, really, so I can't guarantee he didn't find these, and perhaps throw them in the trash, but they aren't in the standard lists of syllogisms I've found, not in first through fourth figures, so he surely, pointlessly, would have. The numbering system is based on the row that is isolated using this row rule method; sometimes there is more than one conclusion that can follow from a given rule.

AaC (5X,7X), BiC (1∨5) ⇒ AiC (1∨3) (R1.1)
AiC (1∨3), BaC (3X,7X) ⇒ BiC (1∨5) (R1.2)
AaC (5X,7X), BoC (3∨7) ⇒ AiC (1∨3) (R3)
AoC (5∨7), BaC (3X,7X) ⇒ BiC (1∨5) (R5.1)
AeC (1X,3X), BiC (1∨5) ⇒ AoC (5∨7) (R5.2)
AoC (5∨7), BeC (1X,5X) ⇒ BoC (3∨7) (R7.1)
AeC (1X,3X), BoC (3∨7) ⇒ AoC (5∨7) (R7.2)

For example, take R7.2, AeC, BoC ⇒ AoC.

Here's how Aristotle's definitions would express it:

If A belongs to no C and B does not belong to some C, then A does not belong to some C.
Wow, that's hard to imagine in my little squirrel brain, I can't even parse that as English. But I'm quite confident of the rule, just by looking at the tables for AeC and BoC, I have no question that if AeC (1X, 3X) is asserted, nothing is in rows 1 or 3, and if BoC is asserted, something is in row 3 or 7, which means that something is for sure in row 7 because it can't be in row 3, and so AoC which is the claim that something is in row 5 or 7 has to be true, because something is in row 7. Make sense?

Okay I'll struggle to make a good example here. Let A be "in my house" and C be "in your house", and B be magic. So AeC says nothing that's in my house is also in your house, which is sensible, since they don't overlap. And BoC says there's a thing somewhere, and we don't know if it's A or not, no comment, but it's not B and it IS C. In this case, it's not magic, and it's in your house. Let's say it's your dog, for example, it's in your house, and it's not magic. Okay, from that can we conclude AoC, that there's a thing somewhere which is not A but it IS C? Your dog is not in my house (not A) but it is in your house (C). So: Yes. Let's refine that to something tighter.

AeC: Nothing in my house is in your house.
BoC: There is something that is not magic that is in your house.
therefore AoC: There is something that is not in my house that is in your house.

So we see that it's true, but B is a sort of distractor. We don't care if B is magic or any category whatever, we just take the assertion of BoC to populate C with something, something that exists that's not B, whatever B might be. Then once C has something in it, whatever else it is or isn't, that's all we needed to have. So given that there's something in C, and C doesn't overlap with A, indeed there must be something outside A that's in C, which is the conclusion AoC. Boom done.

Maybe now is the time to point out a couple errors Aristotle seemed to have made, from a modern perspective. The Third Figure rules, "Darapti" AaC (5X,7X), BaC (3X,7X) ⇒ AiB (1∨2), and "Felapton" AeC (1X,3X), BaC (3X,7X) ⇒ AoB (5∨6), are false, as they stand. Because, in short, ruling out does not populate. There may be special Aristotelian assumptions, like the necessary existence of something in every category, which would make up for these errors. But from our perspective, we won't use them.

Okay enough examples, lengthy perorations, and logical minutiae. What could we use syllogisms for?

Frankly I am very impressed with the language of logical expression that is encoded in Aristotle's symbolic logic with the letters 'a', 'e', 'i', and 'o'. It seems possible to describe any relationships among any number of categories by using this formalism, and further it seems evident to me that an automatic theorem proving system using the valid syllogisms as derivational rules should in principle be buildable to derive conclusions from premises and build valid arguments from simple assertions.

For the purposes of humans learning or teaching or applying logical thinking, however, it seems to me that the syllogistic form, though reparable, enhanced as I hope we have left it now here, is still an increase in noise and a decrease in the signal to noise ratio, for human thought. I believe that humans have to rest often and long within the obvious and undeniable, and then move slowly by easy steps along the paths of ultimately complex reasoning. Staring at incomprehensible formulae with difficult definitions of their parts, I might change careers sooner than persuade anyone else, even if I could persuade myself.

On the other hand the golden simplicity of the possibility table, known in Boolean logic as the truth table, and the intuitive obviousness of a Venn diagram, provide themselves the language of reasoning and logic that we need. All the possibilities are, after all, all the possibilities (H), and when you draw them out and X out the combinations you don't want to allow and populate or universalize the combinations you do want to see, then you can more or less read off the diagram the conclusions that you seek.

So I'm advocating for humans to use tautology and simple, relatively infallible representations, to reason with.

I'll have more to say no doubt, but for now let's consider this enough.

Another Day

Here's what else I have to say.

What have we found?

Let us ignore the irrelevant immaterial non-essentials, the non-essential irrelevancies, and derive a rule-set based on the minimal essentials, thus to see if everything previously derived along with superadditive noise and confusing side specifications can be derived from the essentials. (And indeed so we find.)

So, whether a term is subject or predicate is irrelevant.

Whether a term is "middle" doesn't help us except in sorting the rules in a labelling order, which is immaterial.

Mnemonics are not the point: Validity, truth is the point.

Incidentally the whole terminology of "figures" is just a way to avoid having to say which terms in which order are in any assertion (because knowing the figure tells you it is A, B, and C, in what places in the assertions of a syllogism. This false economy in support of the mnemonics just pulls the wool over our own eyes so we don't see the other valid rules of deduction from the same inventory of original possible primitive statements. Similarly the order of premises is immaterial to the conclusion. From P1,P2 ⇒ C we also have P2,P1 ⇒ C. It's just the reasons for adding C to the proof are taken from wherever, in whatever order, so long as they are there, and validly jointly deduce C.

The Point

This discussion, and Aristotelian logic in general, and logic in general, exemplify Tautology as follows: So, equally valid as Aristotle's four figures of syllogisms (unnecessarily restricted as they are to those in which each of three terms occurs exactly twice in the rule), are additional valid rules of inference with the same vocabulary of connectives or basic assertions, also among three terms in order, namely the seven row rules (R1.1, R1.2, R3, R5.1, R5.2, R7.1, and R7.2) and the double cross out rules (asserting a premised empty set to be a subset of any other set and to have its intersection with any other set to be empty). A modern proof engine could add these Veatch syllogism rules to its inventory of valid deductions as they are no more complex than Aristotle's. Having seen his rules include false ones, Darapti and Felapton, and having seen they fail to include true and valid ones, listed above, we may wonder what is the point of his system.

Surely the point is to have (1) a language for expressing every imaginable relationship between categories, i.e., to describe things, along with (2) a rule system for deriving all valid inferences therefrom. Given categories, and AND and OR and NOT, and perhaps parentheses for priority of evaluation, it seems clear than Aristotle's a,e,i,o can be used to assert any combination of lines in any truth table, so it does satisfy (1). Removing Darapti and Felapton, and adding the 7 row rules and the double cross-out rules (which themselves may be simplifiable), this also satisfies (2). It will do, as a practical theory for a computational logic inference engine.

However, it seems like a lot of memorization and complex machinery. Built upon a sound foundation of tautology as it is, it seems for human purposes that the simpler foundations of AND, OR, NOT, and the truth table concept itself, along with the validity in both form of a statement with a valid form, and in reality of the assertion it makes about reality (which justifies editorial operations from truth table to lines of a proof, and from either to the human intuition of the expressed idea), is all we really need to learn and memorize.

Aristotle, and the 12 valid Aristotelian, and 12 valid Veatch syllogisms can be left in a computer table of rules, or in the homework book, or as homework, to satisfy ourselves that logic is actually all tautology.

Then we can use simple ideas, which are no burden for memory or reasoning, to reason out anything we want to be certain about.

Perhaps, and I only barely say "perhaps", having gone through such exercises we may be better equipped to resolve questions requiring reasoning: such would be the benefit of (classical) education.

On the other hand, if we are fooled by scholastics and logicins into thinking their formulas must be memorized and used for us to have the power of reason, then we will have given away the power of thought, itself, in order to own a spaghetti-cloud of confusion. That's how the pros snow you to get them to do the work: by demonstrating such complexity.

I say urgently, don't be fooled. Keep it simple. Don't be so easily intimidated, so fearful. Trust in tautology. H.

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Copyright © 2022 Thomas C. Veatch. All rights reserved.
Written: February 22, 2022