Answer

Hello.
1. A division into two equal parts does not yield an identity relation. This follows from the fact that although the parts are equal in their properties, they are still two, and replacing one with the other is not an identity operation. According to Leibniz there is a principle of the identity of indiscernibles, but in my book Two Wagons, in the second section, I explained that Leibniz is mistaken. There can be two objects with the same set of properties and they still remain two.
2. In my opinion, a state of love and hate simultaneously is possible. Perhaps the love is directed toward one aspect of the beloved and the hate toward another aspect, but I do not rule out the possibility of love and hate toward the same person even apart from different aspects (the question is whether a relation like love or hate depends on certain aspects or is directed toward the whole).
I did not understand the end of your remarks.
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Questioner:
The question was raised at the beginning of the lecture whether there are relations in reality that are not transitive.
In my humble opinion, there are many –
The relation of group preference, in many of the ways social choice is defined, is not transitive, something known as the “voting paradox.”
The relation “is a friend of” is not necessarily transitive (if Reuven is a friend of Shimon and Shimon is a friend of Levi, we have no way of knowing anything about Reuven and Levi). Similarly, the relation “is an enemy of” is not transitive. Nor is the relation “knows.”
One can define on mathematical objects (or on a row of houses on a street) a relation of “adjacent to…” that is not transitive.
And many more like these.
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Rabbi:
Obviously. Did I say otherwise? (I no longer remember.) The question pertains only to relations that look like order-relations.
The fifth book in the Talmudic Logic series deals with the treatment of non-transitivity.
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Questioner:
 
1. It may be that we are relying on different definitions of the term “identity.” Or, more precisely, I am apparently relying on a very lenient definition that comes from familiarity with the concept in the mathematical context.
As I understand it, the function X+X^2 and the zero function are identical over a field with two elements (where the field is both the domain and the range). This identity follows from my defining a function as a subset of a Cartesian product, etc. (a standard definition, in my opinion).
Of course, if the definition of the function included, for example, its representation “on paper,” there would be no identity here. Likewise, it is clear that if we were discussing polynomials, which are defined by their coefficients, the two examples above would not satisfy an identity relation.
Apparently the Rabbi is referring to the concept of identity in a stricter sense.
I know the principle of the identity of indiscernibles only loosely from a course on symmetry (Buzaglo) that was not entirely mathematical and not entirely philosophical either. In any case, I did not learn there a refutation of this principle (though we did mention, as a basis for the principle, the principle of sufficient reason, and it seems to me that a refutation could come from there). I have not read the Rabbi’s book, and I know his thought from YouTube. Sorry.
 
As for the matter we were discussing—in my humble opinion, even if there is no identity between, for example, “half wise half foolish” and “half foolish half wise,” is there really inversion here? In my opinion, no.
 
 
2. A preference relation over more than two options would be, for example, putting up three candidates for election—Arel, Benjamin, and Gila.
If the public chooses (let us set aside for the moment the question of the voting method, even though it may be relevant to the issue of inversion) the preference vector 0.3, 0.2, 0.5—does it have an inverse? How would it be defined? “Everyone who supported Benjamin now opposes him”? Such an inversion is not well defined, nor does it satisfy properties that it would make sense to require of an inversion, such as uniqueness, or that composing the inversion an even number of times yields the identity.
Would inversion be considered a change in the order of preferences? The problems here would remain as they are.
In my opinion, aside from edge cases (for example, two of the candidates split all the votes between them and the third gets 0), it would be hard to define an inversion here.
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Rabbi:
You lost me.
Everything you wrote seems correct to me, and still I do not see the connection to what I said. Obviously, things discussed in fuzzy or multi-valued logic do not have a unique opposite in the sense of negation. That follows from the very definition of the logic there. What does that have to do with what I said?
The same applies to a preference relation over more than two options. By the way, I dealt with it in almost all my columns on the site (Complex Thinking—mainly in the first column, but in almost all the others as well). But what does that have to do with what I said?