The Five Indemonstrables
The logic that was lost
The Stoics built the first logic of propositions in the West, ran it on five argument forms they held to be complete, and had it forgotten for the better part of two thousand years while Aristotle’s logic of terms was taught in its place. Here are the five, working.
First, what can be true
Logic runs on propositions, and not everything you can say is one. A question is not false. An order is not true. Before anything can be inferred, the Stoics had to say which utterances are even eligible. Sort these.
- It is day.
- Is it day?
- Go thou to the waters of Inachus.
- Most glorious son of Atreus, Agamemnon, lord of men.
- Plato is alive.
- How like to Priam’s sons the cowherd is!
Six utterances. Only some of them can be true.
Interrogations, inquiries and the like are neither true nor false, whereas judgements (or propositions) are always either true or false.
The five, and two impostors
Each form is a shape with two holes in it. Pick a form to see the shape, then drop a pair of propositions in and watch the shape stay exactly as valid, or exactly as broken, as it was. That indifference to content is the whole idea.
If the first, then the second.
The first.
therefore the second.
Valid. modus ponens.
Take the conditional and assert its front half. The back half follows. Everything else in the system is built to be reduced to this.
The first kind of indemonstrable statement is that in which the whole argument is constructed of a hypothetical proposition and the clause with which the hypothetical proposition begins, while the final clause is the conclusion; as e.g. “If the first, then the second; but the first is, therefore the second is.”
A mood is a sort of outline of an argument, like the following: “If the first, then the second; but the first is, therefore the second is.”
Further, arguments may be divided into true and false. The former draw their conclusions by means of true premisses; e.g. “If virtue does good, vice does harm; but virtue does good, therefore vice does harm.”
The three connectives
Set the two propositions true or false and watch what the Stoic connectives do with them. The disjunction is the one that surprises people: the text says outright that “either” guarantees one alternative is false, so it is exclusive, and the fourth indemonstrable depends on that.
The first true, the second false. The conjunction fails and the disjunction holds, exactly one half being true. This is the one case Philo’s conditional is false: front half true, back half false.
Conjunction
the first and the second
True when both halves are true, and in no other case.
Disjunction
either the first or the second
Exclusive, and the text says so outright. Exactly one half is true. This is not the inclusive “or” of a modern logic course, and the fourth indemonstrable depends on the difference.
Conditional
if the first, then the second
The hard one, and the one the schools fought over. This test is not a truth table: it asks whether the denial of the back half could sit alongside the front half, which is a question about what is possible, not about what happens to be the case. The value shown is Philo’s, because his is the only one a table can decide.
It is not the case that Plato is both dead and alive; but he is dead, therefore Plato is not alive.
A disjunctive proposition is one which is constituted such by the disjunctive conjunction “Either,” as e.g. “Either it is day or it is night.” This conjunction guarantees that one or other of the alternatives is false.
A hypothetical proposition is therefore true, if the contradictory of its conclusion is incompatible with its premiss, e.g. “If it is day, it is light.” This is true. For the statement “It is not light,” contradicting the conclusion, is incompatible with the premiss “It is day.”
Three ways to mean “if”
The conditional is where the schools broke apart, and the disagreement is not a quibble: only the first of these can be settled by looking at the truth values. The other two ask what is possible, which no table can show you.
Philo the Megarian
False in one case only: the front half true and the back half false. Otherwise true.
Readable from a truth table
This is the material conditional of modern logic, twenty-two centuries early.
Diodorus Cronus
True when it neither is nor ever was possible for the front half to hold and the back half to fail.
Not readable from a truth table
Philo’s test, run over all of time rather than over this moment. The truth values you can see are not enough to settle it.
Chrysippus
True when the contradictory of the back half is incompatible with the front half.
Not readable from a truth table
The strictest of the three, and the one Diogenes states. It asks for a connection between the halves, not a coincidence of their truth values.
Chrysippus’s is the account Diogenes states, and it is quoted above. Philo’s and Diodorus’s reach us through Sextus Empiricus, who is not in the Corpus, so they are reported here after Benson Mates, Stoic Logic, and are not quoted.
Why you were not taught this
Chrysippus held that every valid argument could be reduced to combinations of these five, and wrote meta-rules for doing the reducing. Most of those rules are lost, so whether he was right cannot now be settled. What can be said is that this is a logic of propositions, where the variables stand for whole sentences, and that Aristotle’s was a logic of terms. For most of two thousand years the second was taught and the first was not, which is why Kant could say logic had not advanced since Aristotle and be believed.
The Impression asks what happens before any of this starts →
Take a side
Chrysippus claimed the five were complete. Was he right, and how would anyone now tell?
Opening the board…