by Bruce E.R. Thompson.
Sometime during the 3rd Century B.C.E. Callimachus, a librarian at the great library of Alexandria, is said to have complained, “The very crows on the roofs croak about the soundness of conditionals!” He was not far from wrong.
Conditional statements are made up of two component statements in which the truth of one (called the antecedent) is a necessary condition for the truth of the other (called the consequent). For example, clouds are necessary for rain (but not sufficient), so, if we know it is raining, we may infer that there are clouds in the sky. The task of logicians is to give elegant and formal descriptions of the ways in which ideas connect together. A description is elegant if it is, as Beau Brummell might describe it, pleasingly simple. The description is formal in that it concerns only the form of the statement, not its specific content. So, formally speaking, conditionals are statements of the form “If P, then Q,”—P and Q being variables that can stand for any content-bearing sentences whatsoever.
Under what conditions are conditional statements true? It seems that this is a question not easily answered. The Roman historian Sextus Empericus—writing some 500 years after the period he was describing—makes clear that there were four distinguishable positions held by the Megarian-Stoic philosophers of ancient Athens. The debate among those philosophers was apparently loud and contentious, prompting poor Callimachus to want to stuff his ears with wax. Over the millennia the debate has not subsided. It rages even today among modern logicians.
The Megarian school of philosophy was founded by Euclid of Megara, not to be confused with the more famous geometer, Euclid of Alexandria. Euclid of Megara was a follower of Socrates, and he lived some 200 years before Alexandria was even built. Euclid of Megara was interested in the Socratic method of disputation, so his school is sometimes called the “dialectical” school. The Megarians were the first logicians to collect examples of fallacious reasoning, including the fallacy we now call “complex question,” which they illustrated using the example, “Have you stopped beating your father?” They were also the first to try to develop the logic of propositions, as opposed to the logic terms, which was developed by Aristotle.
Stoicism, as a school of philosophy, was founded by Zeno of Citium, not to be confused with the more famous creator of paradoxes, Zeno of Elia. Zeno of Citium was strongly influenced by the Megarian dialecticians, and the two schools soon merged into one. Hence, the doctrines concerning disputation that were taught at the Stoa are sometimes called “Megarian-Stoic,” or even simply “Stoic,” even though the doctrines themselves predate the founding of the Stoa.
Sextus describes the four views held by the Megarian-Stoic philosophers concerning conditional statements.
(1) The Philonian position. This view was held by Philo of Megara, not to be confused with the more famous Jewish theologian Philo of Alexandria, who lived around the same time as Jesus.
“Philo said that the conditional is true when it is not the case that it begins with the true and ends with the false.”
Sextus enumerates the logical possibilities, very much in the manner of a truth-table, thus making it crystal clear that Philo means the same thing that modern Boolean logicians mean when they define conditionals truth-functionally: true in all cases, except where the antecedent is true and the consequent false. Thus, according to the Philonian view, the following statement would be regarded as true some of the time:
(a) If it is daylight, then I am disputing with someone about something.
The following two statements would be regarded as true all of the time:
(b) If it is raining, then there are clouds in the sky.
(c) If it is raining, then it is raining.
However, the following (apparently false) statements would also have to be regarded as true all of the time:
(d) If the moon is made of green cheese, then I wear sunglasses when I sleep.
(e) If horses are mammals, then the moon is not made of green cheese.
(f) If the moon is made of green cheese, then the moon is not made of green cheese.
(d) and (f) are true simply because their antecedent is false. (e) is true because its consequent is true. None of them are cases in which the antecedent is true and the consequent false.
(2) The Diodorean position. Diodorus Cronos (who is hard to confuse with anyone else) was Philo’s teacher. Apparently the teacher-pupil relation did not prevent them from having lively disagreements with each other.
“Diodorus says that the conditional is true when it begins with true and neither could nor can end with false. This runs counter to the Philonian position. For the conditional ‘if it is day, I converse’ is true according to Philo, in case it is day and I converse…But according to Diodorus (it is) false. For at a given time it can begin with the true (proposition) ‘it is day’ and end with the false (proposition) ‘I converse’, suppose I should fall silent.”
A strict formulation of the Diodorian position might be: a conditional statement is true if and only if there is no time at which the antecedent is true and the consequent false. Diodorus held the metaphysical view that anything that is possible will, at some time or other, necessarily be actualized. Hence, the Diodorian position appears to be similar to that suggested by modern logicians C. I. Lewis and C. H. Langford, that a conditional strictly so-called might be true when “it is not possible for p to be true and q false.” Lewis and Langford give a rather different account of the concept of possibility than Diodorus does, so the similarity may be only skin deep, but, according to the Diodorian view, the statement,
(a) If it is daylight, then I am disputing with someone about something.
is false all the time, whether it is day or night, and whether I am disputing with someone or not, because it is possible (though unlikely) that I might, on some bright day, find no one to dispute with.
The following would be regarded as true all of the time, since it is not possible for rain to fall without clouds, and it is also not possible for rain to fall without rain:
(b) If it is raining, then there are clouds in the sky.
(c) If it is raining, then it is raining.
However, the following (apparently false) statements would still be regarded as true:
(d) If horses are mammals, then the moon is not made of green cheese.
(e) If the moon is made of green cheese, then the moon is not made of green cheese.
Under the Diodorian view, a conditional is true under the condition that there is no time at which the antecedent is true and the consequent false. But, of course, the moon is never made of green cheese. Hence, whether the antecedent is true (as in (d)) or false (as in (e)), there is no time when the consequent is false, and so there is no time when the consequent is false at the same time that the antecedent is true.
(3) The “connexive” position. Sextus does not attribute this view to a specific spokesperson. However, Diodorus is known to have had five daughters, all of whom were themselves trained dialecticians. At least one of them must have had something to say on this subject, and, since Diodorus could not even get his own pupil to agree with him, there is no reason to suppose that he could get his daughters to agree with him. It would be typical of history to omit the name of a woman. It is only speculation that the original proponent of this view was a woman, but it is a reasonable speculation, and I hope it may be true.
“…and those who introduce the notion of connexion say that a conditional is sound when the contradictory of its consequent is incompatible with its antecedent.”
The historian of logic, I. M. Bochenski, says of the connexivist view, “It is not clear how it is to be understood.” He is right: much turns upon what is meant by “incompatible.” Some modern logicians hold the view that any self-contradictory proposition, such as, “This horse is both white and not white,” is incompatible with any statement—even with itself—on the grounds that it is necessarily false, and so cannot be true in conjunction with any other statement. However, this is the kind of truth-functional thinking that is typical of the Boolean-Philonian school. I think it is more likely that the ancient connexivists took “incompatible” to mean that P and not-Q were not compatible with each other. On this view, P is certainly incompatible with not-P; but, it is not necessarily true that ‘P and not-P’ is incompatible with ‘P and not-P’. After all, what is a self-contradictory statement supposed to be compatible with, if not another self-contradictory statement? According to this interpretation,
(a) If it is daylight, then I am disputing with someone about something.
is false. The claim, “I am not disputing with someone about something,” is perfectly compatible with the claim, “It is daylight.” On the other hand,
(b) If it is raining, then there are clouds in the sky.
(c) If it is raining, then it is raining.
are both true. Rain is incompatible with cloudless skies; and rain is certainly incompatible with lack of rain! The apparently false statements that troubled the previous theories turn out to be false (as they should be) under the connexivist view.
(d) If the moon is made of green cheese, then I wear sunglasses when I sleep.
(e) If horses are mammals, then the moon is not made of green cheese.
(f) If the moon is made of green cheese, then the moon is not made of green cheese.
In each case, there is nothing about the antecedent that would make it incompatible with the denial of the consequent.
(4) The “strict inclusivist” position. Again, Sextus does not attribute this view to a specific spokesperson, so we are free to speculate that the originator of this view was one of Diodorus’ daughters—perhaps the youngest and most obstreperous of the brood.
“Those who judge (implication) by what is implicit say that the conditional is true when its consequent is potentially contained in the antecedent. According to them the (proposition) ‘if it is day, it is day’ and every repetitive conditional is probably false, since nothing can be contained in itself.”
This position differs from the connexivist position in that connexivists accept repetitive conditionals, such as “If it is day, it is day,” as true, since “It is day,” and “It is not day,” are incompatible with each other. Bochenski remarks that the inclusivist position that such statements are not true is “not now fully intelligible.” I’d suggest that perhaps the inclusivists were trying to distinguish trivial statements from true statements. While nearly all modern logicians would say that repetitive conditionals are “true, but trivial,” an inclusivist might say that they are “not true, because they are trivial.” This sounds like an attempt at a four-valued logic in which there are four truth values: trivial, true, false, and incoherent. In any case, while the first three Megarian-Stoic positions have modern apologists, the fourth does not. Multi-valued logics have proven themselves useful for some purposes; but so far, describing the meaning of conditional statements has not been one of them.
It is my considered opinion that the connexivist position does the best job of describing what we mean when we make conditional claims. However, this position also has problems. It lacks an elegant formal semantics.
A formal semantics is a mathematically precise description of the meanings of terms. The Boolean-Philonian position has truth tables, and behind that it is dressed in the elegant formal semantics of classical set theory, as illustrated by the diagrams developed by English mathematician John Venn. The modern versions of the Diodorian position have the glamorous and fascinating “possible worlds” semantics, which are extensions of set theory. But the connexivist position is not yet supplied with a sufficiently elegant formal semantics.
If you, like Callimachus, wish to stuff your ears with wax and hear no more about it, I cannot blame you. But I—one of the crows on the roof—must continue cawing until the debate is settled; and, in my view, the debate will not be settled until the connexivist position is dressed with some degree of formal elegance.