Q&A: Logic
Logic
Question
I read about the logical fallacy called denying the antecedent:
if A then B,
not A, therefore not B.
Does the Talmudic rule “from the positive statement you can infer the negative” not suffer from this basic fallacy?
Answer
There is a dispute in the Talmud whether from the positive statement you can infer the negative, or not. The dispute also appears among the halakhic decisors. On the face of it, one could say that the very reason a double condition is required is precisely because from the positive statement you cannot infer the negative. However, Maimonides distinguishes between these two principles, and this was already noted in Kovetz Shiurim in the name of Rabbi Chaim of Brisk. Hugo Bergmann, in his book Introduction to the Theory of Logic, mentions that one of his students pointed this out to him when they studied the matter from the Talmudic principle of a double condition.
But in truth this is not necessarily connected to the logical issue. The main discussion in the Talmud concerns a person’s intentions. For example, think of someone who stipulates a condition: if you bring me chewing gum, I’ll give you a shekel. Even if he did not state the condition in doubled form, it is hard to say that he must give the shekel in any case. Simply put, from the positive statement you infer the negative. This is not a logically valid inference, but it is a reasonable interpretation of a person’s intent. By the way, even someone who disagrees with this may still agree that this is the person’s intent, but require that the language of the condition be self-demonstrating, meaning that the result should follow from it logically. This is a rule within the laws of conditions, even though he may agree that this is indeed the person’s intent.
Discussion on Answer
I don’t see the difficulty. If conditions are formulated this way in the Torah, one can infer from that that this is the proper formulation of a condition. Just as there are halakhic decisors who learn from Laban that one does not marry off the younger daughter before the elder one. It’s not that Laban was issuing halakhic rulings, but rather there is an assumption here that the description in the Torah’s language teaches something that is inherently correct.
But if it’s only a rule within the laws of conditions, then why did the Torah use this formulation in places where that rule is not relevant? Surely not in order to teach this very principle itself—the Talmud did not think to derive it from there, and if anything, they come as one and therefore do not teach. The move from “a rule within the laws of conditions” to “the proper formulation of a condition” (even when we’re not talking about creating any conditional legal effect) isn’t clear to me. If, when Laban makes a stipulation with Jacob without any oath, “Complete this one’s week and we will give you Rachel as well,” and he doesn’t double it by saying “if you do not complete the week I will not give her,” can one infer from that that there is no need to double the condition?!
If in certain places they write a double condition, that shows that the condition must be doubled. In some places they abbreviated and did not present the full formulation. By the way, that’s also how it is in the Talmud itself, as Tosafot wrote: in a condition formulated as “on condition that,” there is a need to double the condition (unlike Maimonides), and the fact that in the Talmud they do not double it is only because they abbreviated.
As for the issue of two verses, that’s not a question. This is the formulation of a condition, and therefore they wrote it that way. Indeed, it would have been enough to write it only once.
If this is a rule within the laws of conditions, then why does the Talmud in tractate Kiddushin 61a ask about Rabbi Hanina ben Gamliel, who disagrees with Rabbi Meir and holds that a double condition is not required, from a series of verses in which there are conditional promises stated with a double condition, which according to Rabbi Hanina’s view would be superfluous? After all, those are not cases involving the laws of conditions.
The Talmud challenges Rabbi Hanina from the verses: [1] God’s words to Cain, “If you do well, there is uplift; and if you do not do well, sin crouches at the door”; [2] Eliezer’s description of Abraham’s oath to him, “And he made me swear: only to my father’s house shall you go and to my family, and take a wife for my son … then you shall be clear from my oath; if you come to my family and they do not give her to you, then you shall be clear from my oath”; [3] God’s promise to the Jewish people, “If you walk in My statutes … and if you reject My statutes”; [4] the prophet’s words, “If you are willing and obedient … but if you refuse and rebel”; [5] what is said regarding the sota, “If you have not gone astray in impurity under your husband, be cleared … but if you have gone astray”; [6] and from the law regarding one defiled by a corpse, “He shall purify himself with it on the third day and on the seventh day he shall be pure; but if he does not purify himself on the third day and on the seventh day, he shall not become pure.”
In examples 1, 3, 4, 5, the Talmud answers on behalf of Rabbi Hanina that there the negation is, in your terms, contrastive rather than nullifying, and certainly a contrast cannot be inferred. That is, without the doubling we would have thought: if you do not do well, there is neither reward nor punishment; if you reject My statutes, there is neither blessing nor curse; if you refuse and rebel, neither good nor bad; if you have gone astray, neither “be cleared” nor “be choked,” but merely a prohibition.
And in examples 2 and 6, where the negation is nullifying, the Talmud finds an intermediate case: in example 2, an intermediate situation between “they give” and “they do not give,” where the woman wants to come but the family does not; and in example 6, an intermediate situation where he was sprinkled only on the third day or only on the seventh day. From the apparent extra wording of the doubled clause, or from its implication, we understand that in the intermediate case the doubled clause applies (“you shall be clear from my oath,” “he shall not become pure”).