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Rediscovering Talmudic logic

8 June 2009

[The following is based on a tikkun leil Shavuot talk that I gave at my synagogue last week.]

There is a mishnah in the tractate Nazir that I first learned from The Big Book of Jewish Humor, of all places:

excerpt from Nazir 34a. Someone sees a koy [an animal that the rabbis were not sure whether to classify as wild or domesticated] and says

  1. “I swear to be a nazirite [see Numbers 6:1-21] if this is a wild animal”; [others say]
  2. “I swear to be a nazirite if this is not a wild animal”;
  3. “I swear to be a nazirite if this is a domesticated animal”;
  4. “I swear to be a nazirite if this is not a domesticated animal”;
  5. “I swear to be a nazirite if this is both a wild animal and a domesticated animal”;
  6. “I swear to be a nazirite if this is neither a wild animal nor a domesticated animal”;
  7. “I swear to be a nazirite if one of you is a nazirite”;
  8. “I swear to be a nazirite if none of you is a nazirite”;
  9. “I swear to be a nazirite if all of you are nazirites”;

the law is that they are all nazirites.

But the rabbis, it turns out, may have the last laugh.

If this mishnah sounds “illogical” to contemporary ears, it’s because to us, “logic” is a particular system of deduction that we learned in school, a tradition going back to Aristotle—what mathematicians call classical logic. But in the early twentieth century [isn’t it great to talk about “the twentieth century” as an era receding into history?], after classical logic had been reduced to axioms, some mathematicians began experimenting with other axioms, just as a chess master might entertain alternative rule sets for chess. One of these alternatives is intuitionistic logic, which is just like classical logic except that it lacks the law of the excluded middle—the axiom that every proposition must be either true or false.

(If you’re a geek who works with databases you may be familiar with “three-valued logic”, in which Boolean values may be true, false, or NULL. Intuitionistic logic is not three-valued logic. Nor is it fuzzy logic, in which a statement’s truthiness can be measured on a continuum from zero [totally false] to one [totally true]. The axioms of intuitionistic logic neither admit the law of the excluded middle nor define any third alternative to truth and falsehood.)

Let us assume that if you make a conditional oath, the condition is presumed to be true, and if you want to wiggle out of the oath later, you have the burden of proving the condition false. So if P is some statement that can neither be proven nor disproven, “I am a nazirite if P” renders you a nazirite, and so does “I am a nazirite if not-P”. By that principle, the first six oath-takers are uncontroversially nazirites, and there are several ways, depending on your logic system, to prove that the seventh one is as well.

By the time that we get to the eighth man, though, we have to consider what “I am a nazirite if none of you is a nazirite” means. If it means “I am a nazirite if the law would not declare any of you to be nazirites”, then the eighth man is off the hook, because, as we see, all seven of his predecessors end up nazirites. But since the halakhic judgement comes at the very end of the mishnah, maybe we are not allowed to interpret the eighth statement this way; perhaps, at the time the man took an oath, he had no idea what the law would declare, and therefore we cannot recursively apply the law’s judgement to interpret the validity of the oath.

So alternatively, we could interpret it as “I am a nazirite if none of the conditions that you imposed on your oaths are true”. And in order to wriggle out of the oath, the eighth man would have to prove that at least one of those seven conditions are true. Under classical logic, this is easy: “either the koy is wild or not-wild, so either the first or the second condition is true, so hand over the Manishevitz”. But under intuitionistic logic, the eighth man is stuck: he can’t assume “either the koy is wild or not-wild”, so he has to prove which of his predecessors swore by a condition that turned out to be false, which he can’t do.

For a reason that will become clear in the next paragraph, I think the mishnah considers “I am a nazirite if X, who swore another conditional oath, is also a nazirite” to have an ambiguous interpretation, and therefore considers both of the above possibilities and chooses the stricter one. Which is why the eighth man is also rendered a nazirite.

Now let’s consider the ninth. If we interpret “I swear to be a nazirite if all of you are nazirites” as “I swear to be a nazirite if the law would declare all of you to be nazirites”—look, that’s exactly what the law does, and he’s stuck. But if we interpret it as “I swear to be a nazirite if all of the conditions that you imposed on your oaths are true”, then he’s off the hook, because under both intuitionistic and classical logic, a statement cannot simultaneously be true and false, so the koy cannot be simultaneously wild and not-wild. But by the interpretive principle that I suggested in the last paragraph, the law interprets the ninth man’s statement conservatively, and he cannot use the axiom of non-contradition to escape. Thus he, like his eight fellows, is a nazirite.

(Note that none of the characters in our mishnah say “I swear to be a nazirite if this is a wild animal and not a wild animal.” Such a statement would not render you a nazirite, although it would earn you a flogging for the sin making a pointless oath.)

And so, seen through the lens of intuitionistic logic, our mishnah is a little more reasonable, but, I hope, no less funny.

Is this common thread between second-century Jewish law and twentieth-century mathematics a mere coincidence? I think not.

The early intuitionists were not merely tweaking the rules of logic out of whimsy; they had an agenda. Classical mathematics assumes that “the truth is out there” and a mathematician’s job is to discover the timeless properties of some Platonic realm of numbers, triangles, and so on. The intuitionists believed that the truth is in here (he said, tapping his own forehead); mathematical objects are constructed in the mathematician’s mind, and you can only prove things about objects that you can construct.

This attitude has a lot in common with halakhic decision-making. If you have a piece of meat that may or may not be kosher, at a certain point you may need to decide whether or not, in fact, you may eat the meat. If the meat’s status is confined to some ideal realm where all we can do is speculate “there exists in this slaughterhouse at least one side of beef such that…”, it does us no good. Jewish law gives us procedures by which, based on the incomplete information before us, we can make a decision one way or another. And by using these procedures, we are not merely discovering metaphysical facts about the physical world; instead, we create them.