The Brain, a User's Manual:
Matters Syllogistical
By Gabriel Blanchard
Today, we conclude our review of Aristotelian logic and make ready to embark on Stoic logic, the last missing piece of the puzzle.
Alloys
So! We have laid down limits for definition, made our categories substantive, dotted our I and crossed our E statements, and made A square from our O. What is even left to do?
What remains is the syllogism—a word that, like so many words in English, would be self-explanatory if we had built it out of English elements instead of cornering other languages in alleys and mugging them for vocabulary. Anyway, it is derived from two simple Greek elements: σύν [sün], “with,” and λόγος [logos], “reason, computation, order.” A syllogism takes two or more statements and puts them together, to see what can be deduced from them. A statement by itself only means, well, itself; blend two statements together, and you can make something else—a deduction, something you perhaps hadn’t realized before; it’s like smelting tin and copper into bronze, something far harder and sturdier than either element was alone. For example, making use of the handy ∴ sign for “therefore”:
All men are mortal.
Dr. Who is not mortal.
∴ Dr. Who is not a man.
Either that conclusion is true, or something has to be wrong with one of the premises, no two ways about it.
This is what’s meant by validity in logic. If a syllogism is valid, then, by its very structure, putting in true premises necessarily produces a true conclusion—as necessarily as, if A = B and B = C, then A = C too. However, a word of prudence is in order here. One reason logic sometimes gets a bad name is that people who are excited by this guarantee often breezily forget that all-important qualifier, “putting in true premises.” A valid syllogism can only give you what you give it: false premises, or uncertain premises, or “true as far as they go” premises, will only get you a falsehood, an uncertainty, or a conclusion that’s “true enough, depending.” But if we are sufficiently careful with what we mean at each stage in the argument, and true premises are put into a valid syllogism, it will be sound (as in expressions like “soundly constructed” or “home safe and sound”): in other words, its conclusion will be true.
This brings up something we haven’t yet discussed, though we’ve already seen a few examples of it. That something is what’s called a middle term.
Middle Management
Middle terms are fairly simple: they’re the term shared in common by your at-least-two premises. If your premises don’t share a common term …
All men are mortal.
My dog has four legs.
∴ ?
… you can’t really conclude much from that, can you?
But just having a middle term is not always enough. It needs to be in the right place in both statements. That may seem strange at first, but look at it like this. Suppose you think of two generic substances—cats and dogs, say—and realize that they can share the middle term “mammal.” Alright, starting with two A statements:
All cats are mammals.
All dogs are mammals.
∴?
No conclusion follows here; we haven’t learnt anything, only stated facts in sequence. The technical name for how a middle term is positioned is syllogistic figure. There are four figures a two-premise syllogism can be arranged in, and, because of some complicated rules that we don’t need to get into, each syllogistic figure only plays nicely with some pairs of A, I, E, and O statements.1
Using our friends S and P for substances and predicates, and adding an M for middle terms, there are four possible syllogistic figures, shown below. (The formatting has been adjusted a bit, to make the patterns of the terms as clear as possible; the “shape” of the middle terms has been emphasized by putting them in red, and “illustrating” their position above each figure with bars or slashes.)
FIGURE I ( \ )
iiiAll M are P
iiiAll S are M
∴ All S are P
FIGURE III ( | | )
iiiiiiiAll M are P
iiSome M are S
∴ Some S are P
FIGURE II ( | | )
iiNo P are M
iiiAll S are M
∴ No S are P
FIGURE IV ( / )
iiiiiiiAll P are M
iiSome M are S
∴ Some S are P
Man könnte den ganzen Sinn des Buches etwa in die Worte faßen: Was sich überhaupt sagen läßt, läßt sich klar sagen; und wovon man nicht reden kann, darüber muß man schweigen.
One could summarize the message of this book in the following words: Whatever can be said, can be said clearly; and whatever one cannot speak about, one must be silent about.
Mathematically, there are over two hundred and fifty distinct two-premise syllogisms. Thankfully, we don’t need to get into the weeds, because only fifteen of these syllogisms are valid!2 They have been known since the Middle Ages if not before, and were given Latin nicknames in the Scholastic period (the late eleventh to mid-fourteenth centuries). All fifteen are listed below. Each is indicated by the letters of their two premises and conclusion, followed by a hyphen and the number of their syllogistic figure; and just for fun, the nicknames are here too.
AAA-1 (Barbara)
EAE-1 (Celarent)
AII-1 (Darii)
EIO-1 (Ferio)
AII-3 (Datisi)
IAI-3 (Disamis)
EIO-3 (Ferison)
OAO-3 (Bocardo)
EAE-2 (Cesare)
AEE-2 (Camestres)
EIO-2 (Festino)
AOO-2 (Baroco)
AEE-4 (Calemes)
IAI-4 (Dimatis)
EIO-4 (Fresison)
And that, more or less, is that about that! Armed with this knowledge of syllogisms, we are fully two thirds of the way through our planned exploration of logic! It sure is a good thing there won’t be any wrenches thrown in the works in the near future, isn’t it?
Perhaps you couldn’t hear; I said, it sure is a good thing there won’t be any wrenches thrown in the works, isn’t it!
1The reason why the syllogistic figures impose these constraints has to do with a property called distribution: A distributed term refers to all members of its class, categorically. Universal statements (A and E) distribute their subjects, while negative statements (E and O) distribute their predicates—only the I statement on the square of opposition distributes nothing. Though there are other requirements too, the middle term in a syllogism must be distributed at least once for the syllogism to be valid. A syllogism without one commits the fallacy of the undistributed middle, a type of formal fallacy (i.e., its error arises from a faulty argument structure, as contrasted with the informal fallacies or sophistries, most of which fail even to rise to the level of argument).
2Technically, another nine are quasi-valid, for a total of twenty-four. However, of these, some commit another type of formal fallacy (one a little too involved to explain here), and the rest have unnecessarily weak conclusions, so that there is no reason to use them. For example, if all writers are bad-tempered and all bad-tempered people should do penance, then it is technically valid to conclude that some writers should do penance—but the premises also justify the conclusion that all writers should do penance, and there is no logical reason to change the A statement of the conclusion to an I statement (though there could perhaps be a rhetorical one).
Gabriel Blanchard attended Rockbridge Academy, a classical Christian school, and went on to study Classics at the University of Maryland, College Park. He is CLT’s editor at large, and lives in Baltimore, MD.
If you enjoyed this piece, you might also like our series on the Great Conversation—check out our index to its many topics and see what catches your eye. Thank you for reading the Journal.
Published on 12th December, 2024. Page image of Urizen, a character from the mythology of William Blake representing logic, authority, and unmerciful law.