What is Stoic Logic?

And this is what the Stoics are like in logical matters, so they can maintain that the wise man is always a dialectician. For everything is seen through consideration of it in arguments: both what belongs to the topic of physics and again what belongs to ethics.

—From Diogenes Laertius, our major surviving source for Stoic logic.

“Logic” for the ancients included anything that was remotely connected to rational thought. It was the huge, messy toolbox that allowed us to make progress in the the other two branches of philosophy—“physics” and “ethics.”

The Stoics famously compared logic to the wall that surrounds a field (ethics was the fruit, and physics was the land), or to the shell of an egg (with physics and ethics inside).

(Diagram by Massimo Pigliucci)

In its broad sense, then, ancient “logic” included all of what we now call rhetoric, grammar, semantics, logic proper, and epistemology. So Stoic logic involved quite a bit, some of which was extremely influential on later thinkers:

• Stoic rhetoric was similar to other kinds of classical rhetoric—they talked about “forensic” rhetoric, “invention,” and so forth. But the Stoics especially emphasized clarity and brevity in style: saying only exactly what was necessary to get the point across. They also emphasized that rhetoric and moral character go hand-in-hand: you are not a “good rhetorician” unless you are also a “good person.”
• Stoic grammar is pretty much what we now call “classical grammar.” The Stoics were the first Westerners to give names to things like “adverbs,” “conjunctions,” “articles,” “participles,” “passive” verb forms, etc.—their contributions to Western grammar can’t be overstated (though it didn’t hold a candle to Panini’s ancient Sanskrit grammar in India, which anticipated many modern developments 2,500 years before Noam Chomsky).
• Stoic epistemology is rooted in the idea that everything we know ultimately comes to us through our five senses. Sense-perception is the basis of all knowledge. Clearly our senses can lead us astray sometimes, though, and the Stoics endlessly debated how knowledge is possible with their great epistemological rivals, the Skeptics.

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But if we take “logic” in its narrow, modern sense, then Stoic logic—the system of formal logical reasoning developed by the prolific philosopher Chrysippus in the 3rd century B.C.E.—is uncannily similar to the kind of logic you find at the beginning of introductory logic textbooks today. They were way ahead of their time.

Today, almost nobody knows much about Aristotle’s syllogisms, but every computer programmer, digital circuit engineer, mathematician, and philosopher is fluent in something very close to Stoic logic.

This is because the Chrysippus developed the first full-fledged system of propositional logic, which is closely related to modern Boolean logic. If you’ve ever taken an Artificial Intelligence class, a Discrete Mathematics class, or some other general introduction to logic, then you’ve already been exposed to most of Stoic logic’s key components.

In modern terms, what the Stoics provided was a system of natural deduction for propositional logic that is based on five (axiomatic) rules of inference.

There are two parts to a system like this: propositions, and inference rules that act upon them.

PROPOSITIONS

Unlike Aristotle’s more famous (but less powerful) system, which was based on syllogisms, Stoic logic starts with simple propositions—statements that are either true or false. A favorite examples from the ancient texts:

• Dion is walking.

From here, we can build up complex propositions by using logical connectives. The ancient connectives look very similar to what logic students study today, with some light variations:

• Conditional: If it is day, it is light.
• Paraconditional: Since it is day, it is light.
• Conjunction: Both it is day and it is night.
• Disjunction: Either it is day or it is night.
• Causal: Because it is day, it is light.
• The more: It is more day than it is night.
• The less: It is less night than it is day.

Notice that the Stoic or is what we now call an exclusive or. Today’s convention of carefully distinguishing between two types of or didn’t develop until rather recently, so disjunctions were a rather confusing topic for ancient students!

Another topic of debate in the ancient world was how the if statement ought to be interpreted. Today everyone accepts that we should treat an if proposition as “true” even if the antecedent is false. This is called “vacuous truth,” and while it’s a little confusing when you first learn it, it makes a lot of sense with our modern love of treating propositions as Boolean functions and charting their full truth tables:

In the ancient world this was not obvious at all. They had no concept of functions (Boolean or otherwise), and even truth tables would have confused people (“Can’t only one row really be correct in any given situation? How can you tell me that there are four for every if statement?”). So you can be sure that vacuous truth was a tricky thing to discuss.

The Stoics were advocates of the vacuous truth model, and defended it against its critics. Like I said, the Stoics were ahead of their time!

THE FIVE INDEMONSTRABLES

Once you have propositions, you need a way to reason about them. The Stoics called rules of reasoning “indemonstrables,” because they acted as the axioms of the system that everything else was “demonstrated” from.

The Stoics knew that are lots of different “indemonstrables” that we could choose as the basis of our system, but they preferred to use these five (which I will represent in modern symbolic notation):

1. Modus ponens:
   A → B (if A then B).
   A (But A).
   Therefore B.
2. Modes tollens:
   A → B
   ¬B (not B).
   Therefore ¬A.
3. Negated conjunction:
   ¬(A ^ B) (not A and B).
   B.
   Therefore ¬A.
4. Exclusive disjunction with positive:
   A ⊕ B (A exclusive or B).
   A.
   Therefore ¬B.
5. Exclusive disjunction with negative:
   A ⊕ B.
   ¬B.
   Therefore A.

For the Stoics, these rules are the basic building blocks of formal arguments and demonstrations. The difference between the two is that an argument proves something that is rather obvious, while a demonstration starts with obvious premises and leads us someplace unexpected.

Here is one simple example of an argument taken from the ancient texts. It uses the third indemonstrable (negated conjunction):

It is not the case that Plato is dead and Plato is alive.
But Plato is dead.

Therefore, Plato is not alive.

That’s the essence of Stoic logic, but there’s more: the Stoics distinguished between what they called conclusive and non-conclusive arguments, and progressive and non-progressive, possible, impossible, necessary, non-necessary arguments, and so forth.

They also identified what we would call logical fallacies: such as arguments that were “improper by omission,” “logically disconnected,” or “redundant.”

And we know that, to help train people in how to recognize these fallacies and to think through the philosophical foundations of logic, they and their debate partners in other schools dreamed up a lot of subtle little logical puzzles with fancy names like The Master or The Liar.

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But we don’t know much more than that about Stoic logic. This is because almost everything we know about it, and certainly everything in this post, comes from one fragment of an ancient textbook preserved in Diogenes Laertius’s 3rd-century C.E. anthology of philosophy.

The prolific works of Chrysippus, while arguably even more famous and important than Aristotle for the ancients, did not survive the decline of ancient civilization, and it was Aristotle’s logic that dominated logical work in the Latin West.

Logicians didn’t pick up where the Stoics left off and start developing the science of logic to new heights until George Boole and Gottlob Frege picked up propositional logic again in the late 1800’s and launched the modern revolution of mathematical logic, digital circuits, programming languages, Artificial Intelligence, and more.

At the heart of all this, however, are still Chrysippus’s good ol’ propositions. With all that is new, Stoic logic is still doing a lot of the real work in these modern technologies.