I listened to the lectures, and I wanted to ask:
A. Why not remain silent, as you suggested in the first lecture, regarding all metaphysical matters as such?
B. Why can’t one accept certain logical contradictions (perhaps ones that deal with human terms) in metaphysical matters, such as how God is infinite and yet there is still room for created beings, as something legitimate?
The medieval authorities did something similar regarding the question of knowledge and free choice, where they answered that in our terms there is a contradiction, but not in relation to the divine, and that has implications for existence, infinity, etc.

A. Be specific.
B. Because a contradiction says nothing at all (you can infer from it both something and its opposite). And even if the medieval authorities did this, that is because they thought there was no logical contradiction here (in my opinion there is), or because they mistakenly thought one can live with logical contradictions.

A. Why not adopt Wittgenstein’s silence regarding metaphysical matters, not because they are meaningless, but because we do not have the ability to speak about them? Because every metaphysical explanation seems like a certain presumptuous attempt to synchronize terms between us.
B. Accepting the contradiction means that I accept both premises as true even though they seem contradictory to me. For example: I assume that God is infinite for one reason or another, and in addition I assume that there is reality here and that I exist, etc., even though one could say that there is a contradiction here in some sense. One can also understand that this seems contradictory to me because of how I perceive the world, but perhaps in the “metaphysical world” there are more dimensions, and thus there is not really a contradiction. Just as in two dimensions non-parallel lines will necessarily meet, but in three dimensions that is not exactly true… This is similar to Rabbi Shem Tov Gafni’s explanation of why God’s infinity is not harmed by our existence, but to contain this as a “general solution” to contradictions when both premises are strong in metaphysical matters (at least regarding contradictions that arise from human concepts and not internal contradictions, which is already another story…)

A. If you think you have no ability to speak about these subjects (what is the difference between that and saying that such talk is meaningless?), then indeed you should remain silent. I don’t think so.
B. If in your eyes this is a contradiction, even if the truth is that there is no contradiction here, then at least from your own perspective you cannot hold both claims together. Otherwise, you yourself will infer from the system any conclusion whatsoever (after all, logically, from a contradiction one may derive any conclusion, including a thing and its opposite). Therefore it does not matter whether you are right that there is a contradiction or not.
As for Rabbi Shem Tov Gafni, that is a claim intended to resolve the contradiction. There we found a reconciliation of the contradiction, and therefore there is no obstacle to holding both claims together. But as long as you have not found such a resolution, there is no point in it.
Just imagine: can you arrive at the conclusion that God exists and also does not exist, and declare emphatically that perhaps there is no contradiction between those two claims, and continue holding both of them? That is nonsense.

A. There is an essential difference between the physical and the metaphysical in relation to human beings. It seems more plausible to say that because I speak in certain categories that depend on the senses / human intellect, they cannot describe metaphysical concepts—just as it is impossible to explain colors to a blind person or describe a book by S.Y. Agnon through smells. Therefore every attempt to explain the thing in itself, rather than experience / phenomenon, etc., seems generally doomed to failure for a human being. This whole argument relates to the limits of the human intellect and to what it is reasonable to assume, and so it doesn’t seem to me that the “silence” should apply to it. I’d be glad if you would expand on why you do not accept this argument…
B. Accepting the contradiction as legitimate is equivalent to saying that I understand the human intellect to be limited in the metaphysical domain, and from this it is also clear that drawing conclusions from those premises is meaningless for exactly the same reason. When I say contradiction, I mean that there appears to be a contradiction to a person, but not necessarily a contradiction in metaphysical dimensions. If we take the question of knowledge and free choice as an example, it seems that there is a contradiction in how God knows what I am going to do and yet I still have free choice. The obvious answer is that there is not really any contradiction here, and the contradiction comes from our definitions of those concepts—it is in our intellect. For example, in this question our bondage to a perception of time is what causes the contradiction, and when I assume that God is not bound by time, the contradiction is irrelevant to Him and remains only on our side, within our concepts. When I said that there are contradictions, I meant a mistaken perception that projects from human terms onto metaphysical matters, when in truth there is no contradiction. Regarding the example you gave at the end, existence seems such a basic thing that it is tempting to apply it absolutely to everything, but after all, even before Einstein I assume time held a similar status…

A. I already wrote to you that if you do not find any point in talking about something, then don’t talk about it. I didn’t see an argument here.
B. This resolution to the contradiction between knowledge and free choice is nonsense in my view, but that is not the subject here. If there is no contradiction, then there isn’t. But if there is, then you have no option of holding both ends of it. As I already explained, a system that contains a contradiction says nothing at all. Even if the contradiction exists only in your eyes. See perhaps an addition here:

האם אמונה בסתירות לוגיות היא אפשרית?[1]

A. That is a claim you yourself made—you said you brought two explanations for Wittgenstein’s silence in the first lecture on tzimtzum (or at least that is how I understood it; if not, I’d be glad for an explanation of what you did mean).
B. I’ll read the article, God willing. It’s all the same point: in truth there is no contradiction, and if it seems to us that there is, then apparently that is because our concepts are not precise. Why would undermining my ability to formulate precise concepts not remove the contradiction at its root?

Michi, could you expand a bit on what you keep saying, that from a logical contradiction one can derive any conclusion? How can one derive from the contradiction that the table in front of me is both round and triangular the conclusion that my name is Shlomo? Or did I not understand what you meant?

Guy, we’ve completely exhausted this.
Israel, this is a matter of logical formalism, but it is not important for our purposes. It is enough for me that if you accept the claim X and also not-X, then the two claims are not both true—meaning that you do not really accept both of them. Therefore the statement that you accept X has no meaning (and likewise the statement that you accept not-X has no meaning).

Too bad; that very formalism was actually what I wanted to understand. Thanks anyway.

The formalism is not complicated, but the discussion of its meaning is subtle. I’ll try briefly.

A logical proof necessarily takes us from premises to a conclusion. When there is a valid proof, it means that there cannot be a situation in which the premises are true and the conclusion is not (if the premises are true, then the conclusion is necessarily true). This means that a valid proof allows three situations (and not just one):
1. The first is the desirable situation we are aiming at: the premises are true and the conclusion is true.
But the existence of a valid proof does not rule out two other situations (whose existence does not contradict the fact that there is an argument tying the truth of the conclusion to the truth of the premises):
2. The premises are not true and the conclusion is not true.
3. The premises are not true and the conclusion is true.
So we learn that the only situation the proof rules out is:
4. The premises are true and the conclusion is not.

Now think about a situation where the premises contain a contradiction (that is, one contradicts the other). In such a case, it is impossible for all the premises to be true (because if X is true then Y is not, and vice versa). At least one of them is false.
And from this it follows that in such a case the logical proof says nothing at all. Notice possibilities 2-3 above. The existence of a proof allows both of them, meaning that if the premises are not true, you cannot know anything about the conclusion. In other words, one can construct a logical argument that leads us from those premises to any conclusion whatsoever, true or false, and of course also a thing and its opposite.

That is the explanation. This requires clarifying the material meaning of implication from the premises to the conclusion. Logic tends to assume a minimalist meaning of implication, as I described above: that it cannot be the case that the premises are true and the conclusion is not. One can speak about other meanings of implication, ones that require a substantive connection between the premises and the conclusion, but that is no longer within the framework of accepted logic, since implication and the validity of such an argument would then require the content of the claims involved in the inference, and not only their truth values (traditional logic, and certainly the theory of logical inference, deals with form, formalism).

Thank you very much. You taught me something important: in traditional logic, implication is not based on the content of the premise, but on its truth value.

But I can’t quite grasp that: how can one say that the truth of a premise compels the truth of the conclusion (in your words: “it cannot be that the premises are true and the conclusion is not”), if not because of some connection between the contents of the premise and the conclusion?
In the familiar inference, “All people are mortal, and Socrates is a person, therefore Socrates is mortal,” I find a tight connection in the contents of the premises and the conclusion: in the first premise, mortality is linked to humanity; in the second, humanity is linked to Socrates; and in the conclusion, mortality too is found in Socrates, linked through the humanity attributed to him. Without these content-connections, how can one infer the conclusion from the premises?

The example of Socrates is actually a good example. Validity is a formal matter that has nothing to do with content.
Premise: Every X is Y.
Premise: a is X.
Conclusion: a is Y.
I don’t care what the contents are or whether there is any substantive connection between the variables at all—the argument is valid.

By contrast, the argument “every bachelor is unmarried” is not formal. Its validity stems from understanding the concepts involved. You cannot formalize it (represent it formally by variables that do not depend on content).

On the material definition (the one that depends only on truth values) of implication (= entailment), you can read in any logic book.

Thank you for the detailed reply, and I hope I’m not bothering you.

In the sentence “Every X is Y,” the identity relation (“is”) is what connects the contents, and only by its power does the argument become valid.
So I return to my first claim: given contradictory premises, indeed we can get contradictory conclusions, but only ones connected in content to the contents of the premises—not “any conclusion whatever” (that my name is Shlomo)?

By the way, one can formalize the argument “a bachelor is unmarried” like this: “X is not not-X” (more accurately: not-X is not X).

Can you recommend a particular logic book? (I have no background in the field at all.)

Israel, you do not understand what formal logic is (as opposed to content). You really do need to study some logic book. You can use Irving Copi’s book, Introduction to Logic.
It is impossible to formalize “every bachelor is unmarried,” because your formalization already assumes that bachelor is the negation of married. That is where content enters into the formalization.

Thank you.