Susanne Bobzien’s chapter of the Cambridge Companion to the Stoics makes for a tough read, but it is definitely worth it for anyone seriously interested in Stoicism. It covers the basics of Stoic logic, and then some. I will only hit on some of the highlights, skipping much of the more technical details which Bobzien (rightly) gets into.
She begins by reminding her readers that Stoic logic was a type of propositional logic, concerned with what they called “assertibles,” the primary bearers of truth values. This makes perfect sense, given Stoicism’s concern with practical applications of philosophy: after all, human beings communicate by way of propositions, and the goal was to establish under what conditions what we say may or may not be truthful.
For Stoics, so-called “sayables” constitute the meaning underlying everything we say or think. An assertible, then, is a self-complete sayable, though self-complete sayables also include “questions, inquiries, imperativals, oaths, invocations, assertible-likes, puzzlements, curses, and hypotheses.”
Bobzien says that truth and falsehood are properties of assertibles and that “assertibles resemble Fregean propositions in various respects. There are, however, important differences. The most far-reaching one is that truth and falsehood are temporal properties of assertibles. They can belong to an assertible at one time but not at another.” For instance, the assertible “it is day” is true now (because it is, indeed, day), but will be false later on, and true again later still.
The author then gets into a taxonomy of types of assertibles based on ancient sources, starting with simple and moving to complex assertibles, talking about conjunctions, conditionals, etc.
Interestingly: “The Stoic accounts of assertibles reveal many similarities to modern propositional logic, and there can be little doubt that the Stoics attempted to systematize their logic. However, their system is quite different from the propositional calculus. In particular, Stoic logic is a logic of the validity of arguments, not a system of logical theorems or logical truths.”
Bobzien spends a good amount of time explaining in detail how Stoics treated arguments, which were another subclass of complete sayables. “It seems that the criterion for the correctness of the conditional was the Chrysippean one: an argument is valid provided that the contradictory of the conclusion is incompatible with the conjunction of the premisses. Thus, the Stoic concept of validity resembles our modern one.”
Next, Bobzien tackles syllogistic, which was very different, and arguably more sophisticated, than the better known Aristotelian version. Chrysippus distinguished five classes of syllogisms, and later Stoics added two more. I refer the interested reader to the chapter itself, however, as the treatment of this aspect of Stoic logic gets quite complicated very quickly…
How to Be a Stoic 