Logic: Ancient and Modern Syllogisms

If you have ever seen any syllogism, chances are it’s this one:

All men are mortal;
Socrates is a man;
therefore, Socrates is mortal.

Aristotle believed that there were 19 valid syllogisms. Today — since the work of George Boole (and possibly of De Morgan) — we take it that there are only 15 valid syllogisms.

As I have alluded to before, this distinction arises because we consider the possibility of an empty set. This affects the valid conclusions from such statements as “all unicorns are mortal”.

We take that statement to be true, even though we believe that unicorns do not exist; we take it that all zero of them are mortal.

We do not conclude, however, that some unicorn is mortal.

It is exactly that conclusion — an assertion about some thing, as a consequence of assertions about all things — that we reject, but Aristotle accepted.

I will show you these in detail.

Let me remind you that the proof that the square root of two is irrational begins by presuming the contrary: that the square root of two is rational. Since all rational numbers may be written in lowest terms, we conclude that the square root of two may be written in lowest terms. Eventually we show a contradiction: that it cannot be written in lowest terms.

It is crucial to that argument that we reached valid conclusions about something that did not exist.

Let me throw in a comment. Until about 1850, it seems that what we now call traditional logic focused on expressing valid logical arguments as syllogisms. It may be overzealous to say that “logic” was synonymous with “syllogistic reasoning” — but my understanding is that they were trying their best to make it so.

With the advent of symbolic logic (according to the editor’s introduction to Lewis Carroll’s “Symbolic Logic”, bibliography), the focus of logic shifted to: given a collection of premises, exactly what can we say about each of the things discussed?

A marvelous illustration of that goal is all of those puzzles about, for example, five men living in five adjacent houses, painted in five different colors, having five last names, five pets, etc….

So tell me who owns the parakeet?

Along the way you will probably know who owns each pet, lives in which house, etc. We will have determined everything possible about the things presented in the premises.

The editor goes on to say that modern logic appears to be devoted only to the foundations of mathematics; he considers it a shame that modern logic does not investigate such puzzles routinely.

Incidentally, what worked for me in order to solve such puzzles was to lay out several 2-D grids — and just start crossing off impossibilities. I think I played with these puzzles for a couple of weekends once — having bought a few books full of them — but once I had an algorithm for solving them, I put them down.

For me, today, the interesting question would be: how would I write a computer program to generate such puzzles? I’ll confess that it is not a high priority. Far from it, in fact.

Let us now look at the structure of syllogisms.

They have two premises and one conclusion. Each one of those three is one of the four statements we were looking at in the previous logic post:

All U are V;

Some U are V;

No U are V;

Some U are not V.

There is some standard notation associated with those four statements, and we will find it extremely convenient.

In the first place, the two positive assertions

All U are V;

Some U are V;

are signified by “A” and “I” respectively, which are the first two vowels of the Latin “affirmo“. (They also happen to be the first two vowels in “Aristotle”, but we want to think “affirm”.)

The two negative statements

No U are V;

Some U are not V;

are signified by “E” and “O” respectively, which are the two vowels of the Latin “nego“. (They also happen to be the remaining two vowels in “Aristotle”, but we want to be reminded of “negative”, and thence “nego”.)

Our Socrates example, then, could be described as

A
A
A.

(Okay, formally and rigorously we would have to deal with the fact that Socrates may a specific man, but still, we are talking about every one of them. I hope you can be as casual about this as I am; if not, take “man” to be “human being” and replace “Socrates” by “Americans”. The simple fact is, as I said, if you’ve ever seen a syllogism, that was probably it.)

That list of forms is called the mood of the syllogism.

We will find it useful, sometimes, to write those four statements as

A(U,V)

I(U,V)

N(U,V)

O(U,V).

(Sometimes we will only care about the A, E, I ,O and not the specific arguments.)

In addition, like the only example we’ve seen, a syllogism has three terms. First, one of them is common to the two premises; furthermore, that common term does not appear in the conclusion. For our example, the three terms are “man”, “mortal”, and “Socrates”; and our example could be written

A(U,V)
A(W,U)
A(W,V).

The term U, which is common to the premises and which does not appear in the conclusion, is called the middle term. In our example, the middle term was “man”.

The term W, which appears first in the conclusion, is called the minor term; and the term V, which appears second in the conclusion, is called the major term. In our example, “Socrates” is the minor term, and “mortal” is the major.

The premise which contains the major term is called the major premise; the premise which contains the minor term is called the minor premise.

But, instead of denoting things as “major” and “minor” terms, we denote them as subject and predicate.

One of the distinctions between Aristotle himself and subsequent “traditional logic” is the definitions of the subject and the predicate. In traditional logic, we would always write the conclusion as:

S P.

That is, the subject is the minor term; and the predicate is the major term. My preferred terminology for the terms is subject, predicate, middle (S, P, M); and for the premises, simply first and second.

Aristotle wrote it differently: sometimes he wrote

P S.

(He called such syllogisms “inversions of figure 1”.)

Let me back up.

There are four possible ways for the middle term to show up. Nowadays these four arrangements are called figures 1-4.

Our example illustrates figure 1:

M P
S M
S P.

Here are all four figures, and the inversion:

For starters, I will focus on the four figures, and omit the inversions.

The distinction is nothing more than terminology. Write the inversion figure in the first column below… then interchange P and S to get the second column… then interchange the first and second premises to get the third column…

and that 3rd column is precisely figure 4.

It would seem, then, that we can describe a syllogism by its figure and its mood. We can. There is a pseudo-Latin mnemonic for recalling the 19 syllogisms which Aristotle considered valid. The following comes from Gensler (bibliography); it’s the traditional (or medieval) version.

Barbara, Celarent, Darii, Ferioque, prioris;
Cesare, Camestres, Festino, Baroco, secundae;
tertia, Darapti, Disamis, Datisi, Felapton,
Bocardo, Ferison, habet; quarta insuper addit
Bramantip, Camenes, Dimaris, Fesapo, Fresison.

The 19 capitalized names represent the 19 Aristotelian syllogisms. The first three vowels tell us the mood; I believe the specific consonants actually tell how to convert the syllogism to figure 1 and thereby verify it — but we won’t need that.

Incidentally, our example of Socates is an example of “Barbara”, A A A.

What did that silly pseudo-Latin give us? The first line gives us 4 syllogisms in figure 1 (“prioris”). Recall the figure:

M P
S M
S P.

then use the vowels to write out the 4 moods: AAA, EAE, AII, EIO… then write them vertically.

A All M are P \forall x\  Mx \Rightarrow Px
A All S are M \forall x\  Sx \Rightarrow Mx
A All S are P \forall x\  Sx \Rightarrow Px

E no M is P \forall x\  Mx \Rightarrow \neg\ Px
A all S are M \forall x\  Sx \Rightarrow Mx
E no S are P \forall x\  Sx \Rightarrow \neg\ Px

A all M are P \forall x\  Mx \Rightarrow Px
I some S are M \exists x\  Sx \ \land \  Mx
I some S are P \exists x\  Sx \ \land \  Px

E no M are P \forall x\  Mx \Rightarrow \neg\ Px
I some S are M \exists x\  Sx \ \land \  Mx
O some S are not P \exists x\  Sx \ \land \  \neg\ Px

Those all look good to me. There are some formalities to be observed — we really would like to know how to drop and restore quantifiers, so that we can apply all our tautologies, which do not have quantifiers — but if we simply write Barbara as:

\forall x\  (Sx \Rightarrow Mx \Rightarrow Px)\ ,

we certainly want to drop the intermediate term and conclude

\forall x\  Sx \Rightarrow Px\ .

Similarly, Celarent could be written

\forall x\  (Sx \Rightarrow Mx \Rightarrow \neg\ Px)

and we certainly expect to be able to conclude

\forall x\  Sx \Rightarrow \neg\ Px\ .

Darii could be written… okay, I’ve already done this once without ever mentioning it, so this time I’ll point it out as I do it.

We have

\exists x\  Sx \ \land \  Mx\ .

Let y be such an x. (This is called “existential instantiation”, and I’m saving a discussion for the next post. We’ve been doing it automatically since we first started proving things. You might also consider such magical incantations as motivation for the next post.)

Then we have

Sy \ \land \  My\ .

We also have

\forall x\  Mx \Rightarrow Px

but since it’s true of all x, and y is such an x, then we have (“universal instantiation”)

My \Rightarrow Py

and then

Sy \ \land \  My \Rightarrow Py

and then

Sy \ \land \  Py

and then (since y is such an x, “existential generalization”)

\exists x\  Sx \ \land \  Px\

which “proves” Darii. (I expect to write it all out formally in the next post.)

Similarly, for Ferioque we recall

E no M are P \forall x\  Mx \Rightarrow \neg\ Px
I some S are M \exists x\  Sx \ \land \  Mx
O some S are not P \exists x\  Sx \ \land \  \neg\ Px

and we might just recognize that it differs from Darii in having \neg\ Px\ instead of Px\ .

It’s good.

Figure 2? We start by recalling the figure:

P M
S M
S P

and then we recall the mnemonic: Cesare, Camestres, Festino, Baroco, secundae. And then I grab the vowels: EAE, AEE, EIO, AOO… and re-organize it all again.

E no P is M \forall x\  Px \Rightarrow \neg\ Mx
A All S are M \forall x\  Sx \Rightarrow Mx
E no S are P \forall x\  Sx \Rightarrow \neg\ Px

A all P are M \forall x\  Px \Rightarrow Mx
E no S is M \forall x\  Sx \Rightarrow \neg\ Mx
E no S is P \forall x\  Sx \Rightarrow \neg\ Px

E no P is M \forall x\  Px \Rightarrow \neg\ Mx
I some S is M \exists x\  Sx \ \land \  Mx
O some S is not P \exists x\  Sx \ \land \  \neg\ Px

A all P are M \forall x\  Px \Rightarrow Mx
O some S is not M \exists x\  Sx \ \land \  \neg\ Mx
O some S is not P \exists x\  Sx \ \land \  \neg\ Px

I can outline a proof of Cesare as follows.

Rewrite the first premise as

\forall x\  Mx \Rightarrow \neg\ Px

and then

\forall x\  Sx \Rightarrow Mx \Rightarrow \neg\ Px

hence

\forall x\  Sx \Rightarrow \neg\ Px\ .

I’m more than pretty sure we can prove the other three.

Figure 3 will give us problems. We have the figure…

M P
M S
S P

and the mnemonic: tertia, Darapti, Disamis, Datisi, Felapton, Bocardo, Ferison, habet. Just writing out the vowels…

AAI, IAI, AII, EAO, OAO, EIO…

we expect that the first (AAI) and the fourth (EAO) are not valid, because the premises are universal (A or E) while the conclusions assert existence (I or O). But just to see it explicitly, we’ll write them all out.

We write them all out:

A all M are P \forall x\  Mx \Rightarrow Px
A all M are S \forall x\  Mx \Rightarrow Sx
I some S are P \exists x\  Sx \ \land \  Px

I some M are P \exists x\  Mx \ \land \  Px
A all M are S \forall x\  Mx \Rightarrow Sx
I some S are P \exists x\  Sx \ \land \  Px

A all M are P \forall x\  Mx \Rightarrow Px
I some M are S \exists x\  Mx \ \land \  Sx
I some S are P \exists x\  Sx \ \land \  Px

E no M are P \forall x\  Mx \Rightarrow \neg\ Px
A all M are S \forall x\  Mx \Rightarrow Sx
O some S are not P \exists x\  Sx \ \land \  \neg\ Px

O some M are not P \exists x\  Mx \ \land \  \neg\ Px
A all M are S \forall x\  Mx \Rightarrow Sx
O some S are not P \exists x\  Sx \ \land \  \neg\ Px

E no M are P \forall x\  Mx \Rightarrow \neg\ Px
I some M is not S \exists x\  Mx \ \land \  \neg\ Sx
O some S is not P \exists x\  Sx \ \land \  \neg\ Px

We see that we can indeed dismiss the first and fourth (DaraptI and Felapton) out of hand: each has two premises which do not assert existence, but a conclusion that does. Not acceptable today.

To put it another way… in Darapti, for example, choose the sets Px and Sx disjoint, and the set Mx to be empty. Then the premises are true (the empty set Mx is a subset of both Px and Sx) but the conclusion is false.

The four other than DaraptI and Felapton are valid.

Finally, traditionally, we come to figure 4. We recall it:

P M
M S
S P

and its mnemonic: Bramantip, Camenes, Dimaris, Fesapo, Fresison. We should expect that 2 of these 5 are not valid — we’re on the last group and we’ve only ruled out two of Aristotle’s 19. Just selecting out the vowels, we see the two bad ones… AAI (bad), AEE, IAI, EAO (bad), EIO.

But I’m perfectly willing to write out all 5 syllogisms, even though only three of these are considered valid today. (Like all of mathematics, start with some assumptions; ours are just different from Aristotle’s.)

A all P are M \forall x\  Px \Rightarrow Mx
A all M are S \forall x\  Mx \Rightarrow Sx
I some S are P \exists x\  Sx \ \land \  Px

A all P are M \forall x\  Px \Rightarrow Mx
E no M are S \forall x\  Mx \Rightarrow \neg\ Sx
E no S are P \forall x\  Sx \Rightarrow \neg\ Px

I some P are M \exists x\  Px \ \land \  Mx
A all M are S \forall x\  Mx \Rightarrow Sx
I some S are P \exists x\  Sx \ \land \  Px

E no P are M \forall x\  Px \Rightarrow \neg\ Mx
A all M are S \forall x\  Mx \Rightarrow Sx
O some S are not P \exists x\  Sx \ \land \  \neg\ Px

E no P are M \forall x\  Px \Rightarrow \neg\ Mx
I some M are S \exists x\  Mx \ \land \  Sx
O some S are not P \exists x\  Sx \ \land \  \neg\ Px\ .

As before, we reject the two syllogisms that conclude something exists when neither of the premises asserts existence.

Since I grew up with the inversions rather than with figure 4, let me give you the mnemonic for it. This is from Joseph. First, recall the traditional mnemonic:

Barbara, Celarent, Darii, Ferioque, prioris;
Cesare, Camestres, Festino, Baroco, secundae;
tertia, Darapti, Disamis, Datisi, Felapton,
Bocardo, Ferison, habet; quarta insuper addit
Bramantip, Camenes, Dimaris, Fesapo, Fresison.

Then Joseph’s version is

Barbara, Celarent, Darii, Ferio, Baralipton,
Celantes, Dabitis, Fapesmo, Friseomorum
:
Cesare, Camestres, Festino, Baroco; Darapti,
Felapton, Disamis, Datisi, Bocardo, Ferison.

We have replaced the 5 in red by the 5 in bold; and the order is changed among the sets

Darapti, Disamis, Datisi, Felapton, Bocardo, Ferison

Darapti, Felapton, Disamis, Datisi, Bocardo, Ferison

i.e.
Disamis, Datisi, Felapton

Felapton, Disamis, Datisi.

What happened to the vowels? Figure 4 AAI became inversion AAI; AEE became EAE; IAI became AII; EAO became AEO; and EIO became IEO.

Looks plausible to this extent: the conclusions remained the same, and the first and second premises switched. On the other hand, it looks a little implausible because we know we also switched the S and P terms.

So let’s look at the inversions in detail. We have

Baralipton, Celantes, Dabitis, Fapesmo, Friseomorum

i.e. AAI, EAE, AII, AEO, IEO

and the inversion figure is

M P
S M
P S.

On the face of it, two of them are bad. In fact, AAI is the same both ways, and it’s invalid. Let’s look at the other four, and let’s compare them to the corresponding figure 4 syllogisms.

As before, we have to reject AEO because it asserts existence from two premises that do not. Of the other three, we see that — with P and S interchanged — we have the same conclusions. in fact, as I said earlier, if we also interchange the first and second premises, we map figure 4 onto the inversions, and conversely.

The inversions, and figure 4, are different ways of describing the same thing. Interchanging S and P is just interchanging the major and minor premises… just changing which one is major. And it all comes about because Aristotle thought that some premises were more major than others, and just shouldn’t be named minor.

I should emphasize that I’ve shown you Aristotle’s 19 valid, and the modern subset of 15 valid, syllogisms. What I have not done is examine all the possible syllogisms and determine that all the others are, in fact, invalid.

For one thing, it’s not even clear to me whether there are 256 possible syllogisms or 512. Oh, I know the standard answer is 256, but I think I can make a case for 512.

(If the conclusion must be of the form S P, then there are 256: 4 modes (A, E, I, O) for each of three lines (4x4x4 = 64) times 4 figures = 256. But the four figures presume that the third line is SP; if I can write it as PS, don’t I pick up another factor of 2? Yes, they could all be transformed to one of the first 256 by interchanging S and P, but should I count them before I show they’re equivalent to others?)

My preference is to say 512 — but to qualify that immediately: half can be shown equivalent to the other half, so we only need to investigate 256 possibilities.

Actually, we can cut that down quite a bit. I think I’ll show that in the same post in which I discuss universal generalization and its three cohorts.

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