Twenty-four out of two hundred and fifty-six

All men are mortal; Socrates is a man; therefore Socrates is mortal. That is not one of the 256, because Socrates is not a class — but the argument it is usually offered as an example of is, and there are exactly 256 arguments of that shape.

Two hundred and fifty-six cells, twenty-four of them marked. That ratio — about nine per cent — is the first fact worth having, because it says something about the exercise: most arguments of this shape are invalid, and being unable to see why one is invalid is not evidence that it is not.

The four sentence forms #

The system uses four kinds of sentence and no others. Each relates two classes.

A: All $S$ are $P$. Nothing is an $S$ without being a $P$ — the region inside $S$ and outside $P$ is empty.

E: No $S$ is $P$. The overlap is empty.

I: Some $S$ is $P$. The overlap has something in it.

O: Some $S$ is not $P$. The region inside $S$ and outside $P$ has something in it.

Two of these say a region is empty and two say a region is occupied, and the four exhaust the possibilities: universal or particular, affirmative or negative. The letters are mediaeval mnemonics, the affirmative pair taken from affirmo and the negative pair from nego, and they are worth keeping because the whole literature uses them.

Where 256 comes from #

A syllogism has two premises and a conclusion. The conclusion relates a subject $S$ to a predicate $P$; each premise relates one of those to a third term $M$, the middle, which does not appear in the conclusion.

Each of the three sentences is one of four forms, giving $4 × 4 × 4 = 64$ combinations, called moods. And the middle term can sit in four different places across the two premises — subject of both, predicate of both, or one of each way round — giving four figures. So $64 × 4 = 256$ forms in total.

That is the whole space, and its finiteness is the point. Whatever valid means, it can be decided form by form, because there are only 256 of them.

Turning validity into a count #

An argument is valid when the conclusion cannot be false while the premises are true. That quantifies over every possible situation, which is not obviously finite.

The diagram makes it finite. Three classes cut a Venn diagram into eight regions, and every one of the four sentence forms is a claim that a particular set of regions is empty, or that a particular set of regions is not all empty. So a “situation”, as far as these sentences can tell, is nothing but a choice of which of the eight regions are occupied — and there are $2^8 = 256$ such choices.

Validity is therefore: for each of the 256 occupancies, if both premises hold then the conclusion holds. That is $256 × 256$ checks in total, which is a number a person could in principle do by hand and which a machine does instantly. Nothing is estimated and nothing is sampled.

A valid one, and what the picture shows #

The picture is doing the work that the word therefore usually does on trust. The first premise shades; the second places a dot; and the dot lands in a region that is inside $S$ and outside $P$, which is exactly what the conclusion claims. Nowhere was there a choice about where to put the dot: the shading had already emptied everywhere else it could have gone.

That last clause is the mechanism. A particular premise on its own does not say which region is occupied — some $S$ is $M$ allows the thing to be in $S ∧ M ∧ P$ or in $S ∧ M ∧ ¬P$. It is the universal premise, having emptied one of the two, that pins it down. Every valid syllogism with a particular conclusion works this way, which is why no syllogism with two particular premises is valid: without a universal, nothing is emptied, and the dots can always be placed to defeat the conclusion.

An invalid one, and the counterexample #

This one is worth dwelling on because it is the form people actually commit. All cats are mammals; all dogs are mammals; therefore all dogs are cats. Written with real words the absurdity is instant. Written with letters it looks as reasonable as the valid ones — the premises have the same shape, the middle term appears twice, the conclusion joins the two ends.

What is wrong is that $M$ appears as the predicate of both premises, so nothing forces the two subject classes to touch. The mediaeval rule for this is that the middle term must be distributed — used to talk about all of its class — in at least one premise, and here it is not.

That rule and about five others are the traditional way to teach the subject, and the figure above shows what they are shortcuts for. The rules are correct, they are a genuine compression, and they are also exactly the kind of thing that has to be memorised and can be misremembered. The enumeration cannot be misremembered, and it is what produced the picture: the region that decides this case is unshaded, so an occupancy exists that satisfies both premises and refutes the conclusion.

The four figures, and why they are not four subjects #

The middle term can sit in four places, and the tradition treats the four arrangements as four separate chapters. The enumeration shows they are not.

Read the top figure by rows. Rows one to four are the first figure, five to eight the second, and so on. The valid cells are not evenly spread — the first figure has six, the second six, the third six and the fourth six, which is even — but which moods are valid changes completely between them. AAA is valid in the first figure and nowhere else. EIO is valid in all four, and it is the only mood that is valid in all four without the existential assumption — EAO joins it once that assumption is made, and no third mood ever does.

That EIO works everywhere has a reason worth having. Its premises are one universal negative and one particular affirmative, and a negative universal empties a region no matter which way round its two terms are written — no $P$ is $M$ and no $M$ is $P$ are the same claim. So the arrangement of the middle term cannot break it, and it survives every figure.

The mediaeval names encode all of this. Barbara is AAA in the first figure, and the three vowels are the three sentence forms; Celarent is EAE-1, Darii AII-1, Ferio EIO-1, and so on through Cesare, Camestres, Festino, Baroco in the second figure. There are 24 names for the 24 traditionally valid forms, arranged in a verse that generations learned by heart. The consonants are instructions for reducing each form to one of the four first-figure ones, which was the proof technique before there were diagrams: an S means convert the preceding premise, a P means convert and rearrange, an M means swap the premises.

That reduction system is a genuine piece of mathematics and it is doing exactly what a proof theory does — deriving the far cases from a small set of accepted ones. It is also, from where this essay stands, unnecessary. The enumeration does not need the first figure to be privileged, does not need conversion rules, and does not need a verse. It settles all four figures in the same pass, and it produces the same 24.

The nine, and the two-thousand-year disagreement #

Now the interesting part, which is the difference between 15 and 24.

Both premises here say what is empty. Neither says anything exists. So if there are no $S$ at all — if the class is empty — both premises are vacuously true and the conclusion, which claims there is an $S$ that is a $P$, is false.

Aristotle would not have accepted that objection, because in his practice the terms of a syllogism named things: to talk about $S$ at all was to presuppose there were some. Under that reading the argument is valid, and it is one of the traditional 24.

The modern reading takes All $S$ are $P$ at face value as nothing is an $S$ without being a $P$, which is true when there are no $S$. Under that reading the argument is invalid, and the traditional list shrinks to 15.

Nine forms sit in the gap, and the figure at the top shows them as the pale cells. They are exactly the forms whose premises are both universal and whose conclusion is particular. There is no third possibility and no case where the two readings disagree for any other reason — which is itself a small result, and one the enumeration produces for free by running twice.

Neither reading is wrong. They are different systems, and the disagreement is about what a universal statement is for. A classification that says all unicorns have one horn is doing useful work in a modern setting and is nonsense in an Aristotelian one. The modern convention won because it composes better with the rest of mathematics, where empty cases are common and having to check for them every time is a tax.

What the enumeration replaces #

It is worth being explicit about what has just happened, because it is the field’s central move and this is the cleanest instance of it in the collection.

The traditional treatment of syllogisms is a set of rules — the middle must be distributed, two negative premises yield nothing, a negative premise forces a negative conclusion, and so on — each of which is a theorem requiring proof, and which together take a chapter. The rules are then applied to a form to decide it.

What the figure does instead is check every case. There is no rule, no theorem and nothing to remember. There is a finite space of situations, and validity is a property that can be tested against all of them. The chapter of rules becomes about twenty lines of enumeration, and the twenty lines cannot have an exception nobody noticed.

The catch is the one this field always has: the enumeration works because the language was made small on purpose. Four sentence forms, three terms, one middle. Add a fourth term and the diagram has sixteen regions and $2^{16}$ occupancies, which is still finite but no longer a chapter’s worth of anything. Add relations between individuals — every student has a supervisor — and the language leaves the syllogistic entirely, the situations stop being finite, and the two orders of a pair of quantifiers become the whole question.

That is the boundary the syllogistic could not cross, and it is why the subject sat almost unchanged from Aristotle to the nineteenth century. Not because nothing was tried, but because the framework had exactly the expressive power it had, and everything expressible in it had been found.

What a syllogism cannot say #

The last thing worth doing with a complete enumeration is asking what is outside it, and here the answer is short and consequential.

No syllogism has more than three terms. All $A$ are $B$; all $B$ are $C$; all $C$ are $D$; therefore all $A$ are $D$ is a perfectly good argument and is not a syllogism. It is a chain of two of them, and the tradition handles it by chaining — a sorites — which works but is outside the 256.

No syllogism mentions an individual. The Socrates argument in the first line of this essay is not one, because Socrates is a name and the system has only classes. The traditional repair is to treat Socrates as the class whose only member is Socrates, which is a fix and is transparently a fix.

No syllogism relates two things. This is the one that matters. Every student has a supervisor is a statement about pairs, and there is no way to say it with four sentence forms about classes. The system can talk about the class of students and the class of people with supervisors; it cannot express the relationship that makes the second class what it is.

That third limit is why the subject stood still for two thousand years and then moved very fast. Frege’s notation in 1879 allowed quantifiers over individuals and relations between them, and everything above became a small fragment of a much larger system — one in which the number of situations is no longer finite, validity is no longer decidable by exhaustion, and the order of two quantifiers becomes the central difficulty. That is the next essay, and the change of scale between the two is the largest in this field.

What Venn and Boole actually changed #

The diagrams in this essay are usually credited to the wrong century.

Euler used circles for syllogisms in the 1760s and Leibniz had similar drawings a century earlier, so pictures of this kind are old. What is not old is the convention the figures here use: shading for empty, a mark for occupied, and every region present whether or not anything is in it.

That convention is Venn’s, in 1880, and its importance is that it makes the diagram complete rather than illustrative. An Euler-style picture shows one way the classes could be arranged and invites the reader to imagine the others; a Venn diagram shows the fixed template of all eight possibilities, and a premise is a mark on it. Once that is true, checking a syllogism is a mechanical operation on a fixed picture rather than an exercise in imagining hard enough — and mechanical is what makes it checkable.

Boole’s contribution, thirty years earlier, was the algebra underneath: All $S$ are $P$ as the equation $S(1 - P) = 0$, which is the same claim as “that region is empty” and can be manipulated without any picture at all. The two are the same move made twice, and it is worth noticing which one this site is built to prefer. The algebra is easier to compute with and the diagram is easier to be convinced by, and the figures here try to have both: the pictures are drawn from the same enumeration that produced the verdict, so what is convincing and what is computed are the same object.