Q&A: Double Negation
Double Negation
Question
Dear Rabbi Michael Abraham, hello,
As part of studying Rabbi Shmuel Almosnino’s commentary on Rashi on the Torah, we came across the following problem that perhaps you can help solve.
On the verse, “Were there no graves in Egypt” etc. (Exodus 14:11), Rashi says: “Was it because of a lack of graves” etc. Almosnino says:
That is to say, when a negation comes upon a negation, it indicates an affirmative. But here it cannot be interpreted that way, since negation upon negation occurs only when the second is a privative negation, but not with an absolute negation. And the word “there is no” in the holy tongue indicates an absolute negation.
Later on, Almosnino explains Rashi’s interpretation on the basis of the assumption that here indeed we are not dealing with a double negation. We have no difficulty with that.
The difficulty is with the paragraph I quoted above—what do the (logical?) concepts of privative negation and absolute negation mean, and why can the negation of an absolute negation not be interpreted as an affirmative?
Answer
Sorry for the delay (I only just noticed this).
I think he means the old distinction between two kinds of negation: nullifying negation and contrary negation. For example, the nullifying opposite of 1 is 0. The contrary opposite of 1 is -1.
Thus, the relation between cold and heat is contrary, whereas the relation between light and darkness is nullifying. How do I know? Because if you add light to darkness, the result is light; therefore this is 0 and 1. By contrast, heat and cold cancel each other out, and so they stand in a relation of contrary negation [that is, 1 and -1].
Now you can see that after applying nullifying negation to 1, you cannot return to 1 by means of contrary negation (because the contrary negation of 0 is 0 itself), and therefore here a double negation is not an affirmation.
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Questioner (another one):
Maybe he means a different ancient distinction. Double negation is possible only when removing an impediment. “Were there no openings in the gates” would be interpreted to mean that the gate is not blocked but open. And phrasing it as a double negation instead of a direct affirmation comes precisely to emphasize not only that the thing itself exists, but also that there is no obstacle. Here there are three states: no openings, openings that are blocked, and openings that are open. The double negation comes to sharpen that we are speaking about the second and not the third. But with absolute negation there are only two possible states: there is or there is not, and therefore one would not formulate it in the roundabout way of a double negation.
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Rabbi:
I think what you are saying is roughly what I said. I was speaking about the fact that when dealing with negation one must distinguish among three states: 0, 1, and -1, and not just two. Contrary negation moves me between the extremes, whereas nullifying negation moves me from either extreme to the middle state. Therefore the two negations do not cancel each other out. What you suggested, if I understood you correctly, is very similar: an open opening is 1. A blocked opening is -1, and no opening is 0. Right?
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Questioner:
Indeed, on second thought I understand it as you do—that the content coincides. It’s just that the explanation is not purely logical (that the negation of 0 is not -1; because there is language in which the negation of 0 is in fact 1, as in Amos and Lamentations, “darkness and not light”) but interpretive: the extra wording of double negation expresses something different from an affirmation despite the logical equivalence. One negates the negation, meaning one expresses that there is no impediment.
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Rabbi:
Possibly.