Over on youtube, I received the following question:

How the stoic logic avoid the principle ex contradictione quodlibet?

There’s two formulations:

(1)

(2) and(2) it is not a valid sequent in stoic logic, since it requests two distinct sentences, but why (1) it is evidently invalid?

It was a great question, and one too complicated to answer in a youtube comment, so I’m answering it here instead.

The first thing to note in this context is how the Stoics define validity:

A valid argument, according to the Stoics, is an argument such that the negation of its conclusion is incompatible with the conjunction of its premises (Mates, p. 58).

However, Mates goes on to point out that:

This, however, need not be taken as the

definitionof validity but only as a statement of a property which belongs to all valid arguments (Mates, p. 60).

That is, the incompatibility condition is *necessary* for validity, but it need not be sufficient.

So let us consider (1) and (2) more closely: It’s clear that in (1) and (2), the conclusion is incompatible with the conjunction of the premises *only* because the conjunction of the premises (or the single conjunctive premise) itself is already incompatible. So in a sense, the conclusion is not incompatible *with* the premises — it’s nothing about the *conclusion* that is causing the problem. This is our first indication that even though these two sequents might meet the incompatibility criterion, it won’t be enough.

Another way to put it, in modern classical propositional logic, we have Deduction Theorems (for both and ):

In Stoic logic, however, we don’t have the right-to-left direction; as Bobzien notes,

The Stoics considered all conditionals of the form ‘If , ‘ as true (cf. Sextus,

M.8. 281, 466), but ‘arguments’ of the form ‘ are neither axioms of the system, nor can they be derived; the latter is due to the requirement of a plurality of premisses (p. 181).

So this makes explicit why (2) is excluded; it simply doesn’t have enough premises. It also gives us a clue as to how we might be able to exclude (1): It simply doesn’t have the right form. All true (i.e., “sound” in modern vocabulary) arguments are either one of the five undemonstrated arguments, *or* derived from these five via one of the four *themata*. Arguments of the form (1) cannot be so derived. Note that we cannot use the first *thema*, RAA, which says that “If from two [propositions] some third is deduced, either of them with the opposite of the conclusion implies the opposite of the other” (Mueller, p. 201); because we have to start from a successful derivation before we can use RAA to turn it into another derivation.

Does this mean that the Stoic system is incomplete? That their system of argument forms and the rules for putting these argument forms together to derive new, valid arguments does not match up with their conditions for validity? After all, intuitively we have arguments that *look* like they should be valid, but are not derivable in the formal system.

Not necessarily. For if the “condition for validity” is merely necessary and not sufficient, then lacking a sufficient condition for validity, we don’t have a target class of formulas to match our derivation system up with. *If* the sufficient condition is: Incomptability of the negation of the conclusion with the conjunction of the premises *and* derivable from the five undemonstrated arguments, then by definition, this system, if sound, cannot be incomplete.

Alternatively, we can ask what other ways of characterising the conditional could meet the requirement we quote from Mates, but provide a *definition* of validity. Here, it’s worth looking at the variety of implications in play in the Stoic contexts, especially Chrysippus’s, whose “truth-conditions for the conditional were said to involve a connection…this connection was determined indirectly, based on the concept of conflict (maché), and there are some indications that the resulting concept is that of an implication stronger than strict implication” (Bozien, p. 186). If a genuine connection must be established — and by now it is a commonplace for scholars to argue that Chrysippus had a connexive or relevance understanding of implication — between the antecedent and the consequent, then having *merely* incompatible premises will not be enough to establish the incompatibility of the premises *with the conclusion*, hence allowing the Stoics to escape ECQ.

#### References

Bobzien, Susanne. 1996. “Stoic Syllogistic”, *Oxford Studies in Ancient Philosophy* XIV: 133-192.

Mates, Benson. 1953. *Stoic Logic*, University of California Press).

Mueller, Ian. 1979. “The Completeness of Stoic Propositional Logic”, *Notre Dame Journal of Formal Logic* XX, no. 1: 201-215.