by Bruce E. R. Thompson.
“If…then…” statements—or conditional statements—are made up of two component statements. The full conditional statement asserts that the truth of one component statement (called the antecedent) is a sufficient condition for the truth of the other component statement (called the consequent). For example, clouds are necessary for rain, so, if we know it is raining, we presumably have evidence sufficient to assert that there are clouds in the sky. This is common sense. Yet, explaining the conditions under which conditional statements are true is not easy. Look at the following list of conditional statements. Which do you consider true, and which false?
(a) If it is raining, then there are clouds in the sky.
(b) If it is raining, then it is raining.
(c) If it is day, then I am disputing with someone about something.
(d) If it is day, then it is not day.
(e) If the moon is made of green cheese, then I wear sunglasses when I sleep.
(f) If horses are mammals, then the moon is not made of green cheese.
(g) If the moon is made of green cheese, then the moon is not made of green cheese.
(h) If up is down, then down is up.
(i) If roses are red, but roses aren’t red, then roses are red.
The Roman historian Sextus Empericus—writing some 500 years after the period he was describing—tells us that on the question of the exact meaning of conditional statements there were no less than four distinguishable positions held by ancient philosophers. The debate was apparently loud and contentious, prompting Callimachus, a librarian at the Great Library of Alexandria, to complain, “The very crows on the roofs croak about the soundness of conditionals!”
The debate took place primarily within the Megarian school of philosophy, which was founded around 300 B.C.E. by Euclid of Megara (not to be confused with the more famous geometer, Euclid of Alexandria, who lived some 200 years later). Euclid was a follower of Socrates, and he was interested in the Socratic method of disputation, so his school is sometimes called the “dialectical” school. The Megarians were the first philosophers 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 philosophers to develop theories about the logic of propositions. 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, who lived much earlier.) Zeno 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 time are sometimes called “Megarian-Stoic,” or even simply “Stoic,” although none of the doctrines originated with the Stoics.
Sextus describes the four views held by the Megarian-Stoic logicians:
(1) The Philonian position. This view was held by Philo of Megara (not to be confused with the Jewish theologian Philo of Alexandria, who lived around the time of 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 this view, the following two statements are true all the time,
(a) If it is raining, then there are clouds in the sky.
(b) If it is raining, then it is raining.
However, each of these statements are true some of the time,
(c) If it is day, then I am disputing with someone about something.
(d) If it is day, then it is not day.
Both (c) and (d) are true at night, and (c) can even be true during the day in the unlikely event that I can find no one to argue with. Each of the remaining (apparently false) statements are also true, day or night:
(e) If the moon is made of green cheese, then I wear sunglasses when I sleep.
(f) If horses are mammals, then the moon is not made of green cheese.
(g) If the moon is made of green cheese, then the moon is not made of green cheese.
(h) If up is down, then up is down.
Statement (e) is true because its antecedent is false. (I don’t care to divulge the truth-value of the consequent; it is irrelevant in any case.) Statement (f) is true because its consequent is true. Statements (g) and (h) are true for both reasons. In none of these statements is the antecedent true while the consequent is false. (i) is true because its antecedent, “roses are red, but roses aren’t red,” is a contradiction, and therefore always false. But this is a case that deserves special consideration—which we will get to later.
Most people judge only (a), (b), and (h) to be true. The remainder are judged to be either clearly false, or at least highly questionable and probably false. Yet, according to Philo, every single statement on our list is clearly true at least some of the time. Despite this defiance of common sense, the Philonian-Boolean position is currently the received orthodoxy. It is the view taught in modern logic classes and is accepted almost without question among today’s philosophers. Perhaps it is no wonder that many educated people today are suspicious of “reasoning,” “rationality,” and “logic.”
(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.”
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 much like the view proposed by modern logicians C. I. Lewis and C. H. Langford that a conditional is true when “it is not possible for p to be true and q false.” This is an improvement on the Philonian position in that it allows us to declare (c) and (d) false, but statements (e), (f), (g), (h), and (i) must still be regarded as true. Horses are always mammals, the moon is never made of green cheese, up is never down, and roses are never both red and not red simultaneously. For these statements, there is never a time when the consequent is false and the antecedent true.
(3) The “connexive” position, so named because of the typically British spelling used by classics scholar Martha Kneale in her translation of Sextus. 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 either. 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 is 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.” The question is, “What is meant by incompatible?” Some modern logicians hold that any self-contradictory proposition, such as, “Up is down,” 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. But this is the kind of truth-functional thinking that is typical of Philonians. The ancient connexivists probably meant that the antecedent and the denial of the consequent were not compatible with each other. On this view, (a) and (b) above are both true (as they should be). Rain is incompatible (for causal reasons) with cloudless skies; and rain is certainly incompatible with lack of rain! Moreover, the apparently false statements (d), (e), (f), and (g) that troubled the previous theories also turn out to be false (as they should be) under the connexivist view. In each case, there is nothing about the denial of the consequent that is incompatible with the antecedent. That is genuine progress! What about (h)? Presumably connexivists regarded it as true. After all, what is a self-contradictory statement supposed to be compatible with, if not with itself, or another self-contradictory statement? And finally, what about (i)? In this case the antecedent makes two distinct claims that contradict each other, but the consequent is not self-contradictory, and the denial of the consequent is also not self-contradictory. I believe the ancient connexivists would have judged this statement to be false on the grounds that the denial of the consequent is consistent with at least one of the two assertions in the antecedent and might even be consistent with the two of them put together. But this is speculation. The real question is: what do you think?
(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’ other 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.”
Nearly all modern logicians, including connexivists, would say that repetitive conditionals such as (b) and (h) are true, but trivial. Perhaps the inclusivists were trying to distinguish trivial statements from true statements: these statements are not true because they are trivial. This sounds like an attempt at a logic in which there are four truth values: trivial, true, false, and incoherent. While such distinctions can be drawn, the need does not seem urgent, and this tells us nothing about what an inclusivist would think in each of the other cases.
In my opinion the connexivist position does the best job of describing what we mean when we make conditional claims in common speech. In fact, in ancient Greece, the connexivists effectively won the debate: Aristotle was a connexivist, and the principles of the Aristotelian syllogism are connexive principles. Later Stoics such as Chrysippus also endorsed connexivism. Why, then, did the ancient Megarians propose all those other theories? And why today has connexivism been rejected by all but a few modern logicians? The problem has to do with statement (i) above. That statement presents us with a conundrum.
If you take another look at statement (i) you will see that it follows a form called Conjunctive Simplification. This is the principle that if two statements are true together, then each statement must also be true by itself. For example, if there is a knife on the table and a fork on the table, then there is a knife on the table. This principle seems so obviously true that modern logicians treat it as sacrosanct. It is a basic law of logic.
But is it? While connexivists will admit that Conjunctive Simplification is usually correct, they deny—and, for the sake of consistency, must deny—that it is always correct. There are exceptions, and statement (i) is one of them. Modern logicians have used set-theoretic models to prove that connexive logic is fundamentally inconsistent with the principle of Conjunctive Simplification. You can be a connexivist, or you can believe that all instances of Conjunctive Simplification are valid. You can’t have both.
If you, like Callimachus, wish to stuff your ears with wax and hear no more about it, I cannot blame you. But, let me tell you why I—one of the crows on the roof—think this is important, and therefore must continue to caw.
The future is not yet a fact. The future does not yet “exist.” Hence, any statement we make about the future, or about the world of imaginary beings, or about the world as we imagine it might become, is—by the principles of Philonian-Boolean logic—false. The result is that—by the Philonian-Boolean interpretation of ‘if…then…’—rational moral judgments about how our actions will affect the future are impossible. Speculative fiction—or what our editor Tod Davies calls visionary fiction—is meaningless, trivial, and uninstructive. We can’t seriously consider the consequences of our suppositions because all suppositions have the same consequences namely everything. Once we posit a proposition that is not true, anything whatsoever follows from it. That, of course, is why the Philonian-Boolean interpretation is not merely wrong (in the sense that it does not comport with common sense) but is insidiously destructive to our ability to engage in consequentialist moral reasoning, formulate scientific hypotheses for future testing, or to engage in constructive imagining about an as yet non-existent future.
By contrast, the connexive interpretation comports with common sense and it is NOT insidiously destructive to futuristic reasoning. That is why we should reject – not rationality itself – but the bizarre interpretation of rationality currently taught in most logic classes (although, I hasten to say, not in mine!) by Boolean-Philonian logicians.
I want to be honest, so I must explain that there are real reasons why so many modern logicians prefer the Philonian interpretation. Those reasons (on the surface) have to do with the principle of Conjunctive Simplification, but the reasons run deeper than that. The principles of Boolean logic are founded in classical set theory, creating a grand synthesis of logic and mathematics. That was Bertrand Russell’s contribution to philosophy, and he is rightly famous for it. Aristotelian syllogistic logic (a form of connexivism) does not fit into classical set theory. It never did. That is why Boole (and his followers) rejected Aristotelian syllogistic logic in favor of a stripped-down theory of syllogisms that recognizes fewer valid syllogisms than the Aristotelian system does.
But all is not lost. Set theory can be expanded. It has recently been expanded to include “fuzzy” membership values so that elements can partly belong and partly not belong to sets. This expansion has been hugely successful, but it is still not enough. Aristotelian logic still does not fit within it. Even so, further expansions of set theory remain possible. There is one, which I call bipolar fuzzy sets, into which Aristotelian logic—and by extension, connexivism in general—does fit. A characteristic of bipolar fuzzy sets is that, in bipolar fuzzy sets, it is NOT the case that every set contains the empty set. Hence it is not the case that every proposition follows from a nullity (or falsehood). The details of bipolar fuzzy sets have not yet been fully worked out, but when (or if) they are, we connexivist crows may yet caw the Philonian crows into silence.
In a Facebook post, Tod once expressed the hope that some future logician might someday find a way to reconcile human rationality with the human imagination. Connexive logic tries to do just that. If we can understand what we mean when we say “if…,” we can understand what we mean when we say “what if…?”