Much of our last episode dealt with what Aristotle meant by words like ‘every’ and ‘some.’  As we discussed at some length in our previous post, in the Aristotelian setting, the meaning of ‘every’ was slightly different from what we’re used to.  Under today’s meaning of the word ‘every,’ when I say ‘every frog is green,’ you can check to see whether what I just said is true by checking to see whether the set of frogs is a subset of the set of green things.  But in Aristotle’s philosophy, it takes more to make the sentence ‘every frog is green’ true.  It’s not enough for the set of green things just to contain the set of frogs.

What more does it take to make ‘every frog is green’ true?  The bad news is that we don’t really know.  But the good news is that we do know this: in order for it to be true, ‘frog’ has to be a substance property, and ‘green’ has to be an accidental property.  (See the previous post for an explanation of what these terms mean.)

In our conversation with him, Marko Malink observes that this subtle distinction between what we mean by ‘every’ and what Aristotle meant by ‘every’ doesn’t make much of a difference most of the time.  But it does make a difference when we’re talking about Aristotle’s modal syllogistic.  A huge difference, in fact.

Let’s back up a bit.  Modal logic is the study of reasoning about possibility and necessity.  Possibility and necessity may sound like abstruse academic terms, but in fact we spend nearly every waking moment thinking about what’s possible or necessary, regardless of whether we’re in the classroom.  For example, when making a stew, before I put the carrots, onions, celery, and beef into my stockpot, I’ll first want to consider whether it’s possible to fit them in there, or whether I need a bigger pot.  Before I run to catch the bus down the street, I’ll first want to consider whether it’s possible for me to make it there before the bus drives off.  And before I promise to help my friend move his couch bed up four flights of stairs, I’ll first want to make sure that it’s possible for it to fit through the front door.

Those are examples of physical possibility, which is interesting in its own right.  But philosophers have traditionally been interested in possibility in the very strongest sense.  Something like: a situation is possible just in case it’s conceivable.  If you can coherently imagine it, then it’s possible.  You might call this logical possibility, as opposed to physical possibility.  For example, it’s logically possible for all the cars parked on the street to start floating in the air.  Why?  Well, just picture it to yourself, and you’ll see.  Though it’s never going to happen, there’s nothing preventing you from imagining it.  Now contrast that case with a logical impossibility: a giraffe that’s taller than itself.  You can’t so much as begin to picture that scenario to yourself, because it isn’t coherent.

Let’s take a look at a valid modal argument:

It is possible that my car will randomly start floating in the air.
Therefore, it is not necessarily the case that my car won’t randomly start floating in the air.

(If you don’t know what a valid argument is, take a look here for a definition.)  This is a valid argument, because if the premise is true, then the conclusion absolutely must be true.  There is no way for the former to obtain without the latter also obtaining.

Anyway, that’s a tiny taste of modal logic.  So how about Aristotle?  Let’s not forget him.  Just as Aristotle’s syllogistic was the first logical framework ever to capture any kind of reasoning, his modal syllogistic was the first logical framework ever to capture reasoning about possibility and necessity.  Central to both the regular syllogistic and the modal syllogistic were Aristotle’s lists of all the valid inference patterns he could think of.  A valid inference pattern is a schema for a good argument, which you make by generalizing some particular good argument as much as you can–the goal is to say that any argument that has the resulting form will be valid.  So to make the above valid argument into a valid inference pattern, we’d abstract away all talk about cars floating in the air:

Possibly, A is the case.
Therefore, A isn’t necessarily not the case.

As it turns out, the converse is also a valid pattern of inference:

Necessarily, A is the case.
Therefore, it isn’t possible for A not to be the case.

The idea here is that no matter what sentence you plug in for ‘A,’ you’ll get a valid argument.  This rule of inference will come in handy later on, so let’s call it the principle of duality.  Aristotle’s work was chock full of these sorts of things, and the inference patterns in his ordinary syllogistic all seemed correct.  This was one of the reasons it was so influential–it was the standard reference on logic for thousands of years!  But the modal inference patterns he provided, on the other hand, have always seemed unintuitive.  Even his student Theophrastus was dubious about this one:

Every A is a B.
Every B necessarily is a C.
Therefore, every A necessarily is a C.

According to Theophrastus, this pattern of inference is invalid: the premises can be true, while the conclusion is simultaneously false.  To illustrate this, he gives an example of a bad argument that has that form:

Everything that walks upright is a man.
Every man is necessarily an animal.
Therefore, everything that walks upright is necessarily an animal.

To see that this argument is invalid, consider the following counterexample.  In Ancient Greece, it was true that only men walked upright.  And it was also true (as it is today) that every man was necessarily an animal.  Try to think of a man that isn’t an animal, and it’s kind of like trying to think of a giraffe that’s taller than itself–it just doesn’t make sense!  However, even though these premises were true, the conclusion was false.  Why?  Well, by the principle of duality, the conclusion of this argument is saying that it’s impossible for something that walks upright not to be an animal.  But this wasn’t true in ancient Greece.  A robot can walk upright, even though it isn’t an animal.  And although they didn’t actually have robots in ancient Greece, it was perfectly imaginable that there might be such a thing.  Imagining something that isn’t an animal walking upright makes sense, unlike imagining a man that somehow isn’t an animal, or a giraffe that’s somehow taller than itself.

This is where Marko Malink’s discovery about what Aristotle meant by ‘every’ really proves useful.  Theophrastus accused Aristotle of making a bad prediction: of incorrectly telling us that the above argument ought to be valid.  But actually, Malink argues, Aristotle’s theory made the correct prediction.  Aristotle wouldn’t have thought that the above argument exemplified his pattern of inference, because of his rule that sentences of the form ‘Every A is a B’ can only be true if A is a substance property.  Since walking upright is an accidental property (as we learned from the robots), Aristotle would have granted that every man walks upright, but not that ‘everything’ that walks upright is a man.

For a long time, folks have scratched their heads over how the inventor of logic could have gotten things so right when talking about plain old ordinary logic, and yet so wrong when talking about modal logic.  But now, it would seem that Malink has cracked the problem: for if you restrict yourself to sentences in which ‘every’ obeys the rule just stated, then it turns out that the patterns of inference that Aristotle discusses in his modal syllogistic work just fine!

Matt Teichman