Self-Reference – Lesson 1

Table of Contents

• The origin of the series and the structure of the discussion
• Defining self-reference and reflexive relations
• Self-reference of propositions and the liar paradox
• Pure self-reference and the unavoidable paradox
• Additional paradoxes of self-reference
• A non-necessary link between self-reference and paradoxicality
• Non-paradoxical self-reference and the anti-paradox
• Bertrand Russell's theory of types and its critique
• Three-valued logic as a critique of changing the definition
• Human self-reference and the researcher as part of what is being studied
• Shalom Hanoch: "A person lives inside himself" as self-estrangement
• Shalom Hanoch: "A Song Without a Name" and a thing referring to itself without a name
• Interim summary and the planned continuation

Summary

General overview

The motivation for the series grows out of the conclusion of the "Conceptual Analysis" series at the beginning of the month of Av, where the topic of two forces within the human being came up and was supposed to lead into a discussion of self-reference and self-instruction. The speaker decides to turn that endpoint into a new series and expand it, alongside columns he wrote on the site about the topic. He opens with a general conceptual-philosophical discussion and then plans to examine applications in the field of Jewish law. He defines self-reference as a situation in which a relation that is usually defined between two different things is applied when the second thing is the very same thing itself. He presents the connection between self-reference and paradoxes as a possible but not necessary circularity, critiques "solutions" such as Russell's theory of types and three-valued logic as evasions or changes of definition rather than real solutions, and beyond logical propositions he describes self-reference in human beings as a structure in which the subject is forced to estrange himself into an object. He illustrates this through Shalom Hanoch's songs "A person lives inside himself" and "For my song is an echo in the wind," in which self-reference appears in a way that generates estrangement and the absence of a "name" when a thing describes itself.

The origin of the series and the structure of the discussion

The speaker states that the new series grows out of the final topic he did not manage to complete in the previous series at the beginning of the month of Av, and that topic was supposed to open the way to a discussion of self-reference and self-instruction. He notes that he wrote a column on the site about self-reference and that another column on the subject is about to appear, suggesting that they be viewed as complementary context, while in the lectures he expands much more extensively. He declares that he is beginning with a general conceptual-philosophical discussion and only at the end will he examine applications of the ideas in the realm of Jewish law.

Defining self-reference and reflexive relations

The speaker defines self-reference or self-instruction as a situation in which a sentence or a person refers to itself, and distinguishes between self-reference of propositions and self-reference of human beings. He illustrates a two-place relation with examples like "x is the father of y" or "Reuven is taller than Shimon," and explains that self-reference arises when, instead of placing y, you place x itself. As an example he raises the question whether a person is "his own brother" according to a definition of brotherhood as "having the same parents," and describes how formally the relation can be considered reflexive even though it feels strange when a relation that assumes two different objects is applied to the same object.

Self-reference of propositions and the liar paradox

The speaker presents the liar paradox as a central example of self-reference, and brings the version "All Cretans are liars," whose source is in the New Testament. He argues that this formulation is not a real paradox, because the negation of "All Cretans are liars" is "It is not true that all Cretans are liars," that is, "There is at least one Cretan who is not a liar." That one does not have to be the speaker himself, and therefore the loop can stop. He attributes to Maimonides in Milot HaHigayon this logical distinction, using "Boethius's square of opposition," according to which the negation of a universal affirmative statement is a particular negative statement.

Pure self-reference and the unavoidable paradox

The speaker explains that in order to get a paradoxical loop that you cannot escape from, you need a sentence that refers only to itself without hidden generalizations, such as "Sentence A: Sentence A is false." He shows that formulations like "I am a liar" still contain a hidden generalization ("All my statements are false"), and therefore their negation yields only the existence of one statement that is not false and does not force the statement under discussion to be that exception. He concludes that "pure" self-reference is especially prone to circularity, because it does not rely on a broader set that allows the loop to stop.

Additional paradoxes of self-reference

The speaker brings up "the barber from Seville," who shaves all the people who do not shave themselves, and asks whether the barber "shaves himself," showing that the barber's policy together with self-reference leads to the loop of "if yes then no, and if no then yes." He also presents the paradox of Anaxagoras's contract with the student regarding payment of tuition depending on winning or losing the first trial, and shows that the first trial itself becomes the arena of the condition and therefore a loop is created. He presents Russell's paradox in set theory through "the set of all sets that do not contain themselves as a member," and the question of whether that set includes itself, showing the same entanglement of self-inclusion that gives rise to contradiction.

A non-necessary link between self-reference and paradoxicality

The speaker qualifies the point by saying that self-reference is not identical with paradoxicality, that there are self-references that are not paradoxical, and also paradoxes that do not depend on self-reference. He presents the "Swedish army paradox" of the "surprise drill" as a case where backward reasoning eliminates every possible day of the week even though life experience allows for a surprise drill, and he describes it as a paradox that is not a truth-falsehood loop and does not depend on something referring to itself. He also presents the paradox of "the smallest number that can be described in fewer than a thousand letters," where the very wording creates a short description of a number defined as not describable briefly, and he identifies circularity there as well even without direct self-reference.

Non-paradoxical self-reference and the anti-paradox

The speaker gives examples of self-reference that creates no problem, such as "All sentences are made of words," including the sentence itself, or "This sentence is made of words" as a pure self-reference that is not paradoxical. He distinguishes between a paradoxical sentence that has no consistent truth value and a sentence like "This sentence is true," which, in his view, can be consistent whether you assume it is true or whether you assume it is false, and therefore he calls it an "anti-paradox." He states that an ordinary sentence receives only one truth value, whereas an "anti-paradox" receives two possible truth values, and uses this to demonstrate that self-reference does not necessarily entail a liar-type paradox.

Bertrand Russell's theory of types and its critique

The speaker describes Bertrand Russell and Whitehead and Principia Mathematica, and presents the theory of types as a hierarchy in which each proposition belongs to a certain type, with the rule being that a proposition cannot refer to propositions of its own type but only to lower types. He explains that when such a rule is adopted, self-references are considered "illegal," and therefore the paradoxes will not appear in the language. He accepts the distinction that this can be useful as a mathematical language that prevents paradoxes from appearing, but argues that this is not a solution to paradoxes but a prohibition against expressing them, comparing it to Stalin's method of "solving problems" by removing whoever raises them. He adds that this theory also throws out of the language perfectly valid sentences that are not paradoxical, such as true sentences involving self-reference, and therefore there is no justification for the prohibition if the whole point is just convenience or avoiding complications.

Three-valued logic as a critique of changing the definition

The speaker describes proposals for three-valued logic in which every proposition has three truth values: true, false, and "paradox" (P), and presents this as a way of attaching a "truth value" to paradoxes. He argues that this resembles the previous solutions in that it "stretches the question mark and turns it into an exclamation mark"—that is, it gives the problem a name instead of justifying the third possibility or explaining it. He concludes that changing the definitions of a language or of the mechanism of truth values does not solve a problem, but only changes the way it is expressed or pushes it into a different framework.

Human self-reference and the researcher as part of what is being studied

The speaker moves on to human self-reference and distinguishes it from the logical plane of propositions, arguing that it does not have to be paradoxical in the sense of truth values. He gives the example of studying logic or the intellect, in which a person uses his tools of thought in order to investigate those very tools of thought—in other words, "the intellect investigates itself." He presents the possibility of discomfort or limitation in the fact that there is no point of view "from outside," but argues that there is no logical necessity that such conclusions are incorrect; at most, the research may be partial.

Shalom Hanoch: "A person lives inside himself" as self-estrangement

The speaker describes how he arrived at the connection between two columns he wrote about Shalom Hanoch's songs, "A person lives inside himself" and "For my song is an echo in the wind," and identifies in them the same kind of thinking about self-reference. He quotes from the song "A person lives inside himself" the tension between closing and opening a door, and the line "But most of the time a person is a stranger even to himself," and interprets this as self-reference in which the relation "lives inside" receives the same x as y. He presents the idea that in order to refer to myself I am forced to grasp myself as other, as a subject facing an object, and therefore self-reference generates estrangement, while the ability "to look at myself from outside" is the ability of estrangement.

Shalom Hanoch: "A Song Without a Name" and a thing referring to itself without a name

The speaker quotes from "For my song is an echo in the wind" and explains that the song speaks about itself—meaning, "this song is this very song itself"—and therefore it is a "song without a name" in the sense that a name comes from outside and is given by another. He explains that in his view a name is a way in which the environment relates to you, and when a thing relates to itself it has no name because it has no external point of view from which to be given a name. He connects this to the idea that in the relation between the observer and the object, self-reference requires constructing a foreign model or an imagined outside position; otherwise, "it isn't reference," because reference requires two.

Interim summary and the planned continuation

The speaker sums up by saying that he has shown self-reference of propositions, including cases that generate paradoxes and cases that do not, and then moved on to self-reference of human beings as a cognitive phenomenon in which the same person is both the observer and the object of observation. He states that the problems in the human realm are not identical to the logical problems of truth values, but they may resemble them in certain respects, and he intends to deal with that further on. He concludes with the blessing, "Shabbat shalom, everyone, Shabbat shalom, all the best."

Okay. Actually, the motivation for this series came up for me because at the end of the previous term, at the beginning of the month of Av, we finished the series on conceptual analysis, and the last topic I wanted to deal with and didn’t get to—I started it but didn’t get to it—actually had to do with two powers within one person, which was leading us, or was supposed to lead us, into a discussion of self-reference, self-direction, self-reference. Since we wanted to begin a new series after the break and not just finish the previous one, I simply decided to take that stopping point and turn it into a series, expand it and make it into a series. And the truth is that only this morning, or maybe yesterday, I don’t even remember, I recalled that on the website I actually wrote a column now about self-reference, and in a little while the second column on that topic will come out too, and I had completely forgotten that this was the topic I was going to talk about in these classes. Of course, it was probably somewhere in my head, so it’s not by accident that I decided to deal with exactly this, but I had even forgotten that connection. In any case, the last column and the next one—I hope it goes up tomorrow as well—also deal with this topic, so you can look there a bit too. Here, of course, I’ll expand much more.

All right, so the topic, as I said, is self-reference, and as is my way in these series, I’ll begin with a general conceptual-philosophical discussion, and afterward, in the end, I’ll examine applications of these general ideas in the realm of Jewish law. So what exactly is self-reference or self-direction, self-reference, right? It’s a situation in which a statement or a person—I’ll distinguish between those two in a moment—refers to itself. Yes, the liar paradox is perhaps a well-known and clear example of the matter: a statement that talks about itself and says, “This statement is false,” I, right, this very statement is false. That is self-reference, because the statement refers not to something outside itself, but to itself.

Maybe to sharpen it a bit more, I’ll put it like this. Reference usually, relation usually, is between two different things, right? A relates to B, one person relates to another object, one statement refers to a statement, to an object, to another situation. So a relation is usually—or what’s called a two-place relation, a two-place predicate—usually marked… you know what, maybe I’ll do screen sharing. One second, I’ll just write it and it’ll be easier for me to show. What’s going on here? Ah, there. Wait, screen sharing. One second, it refuses to let me share here, not clear to me why. I’m trying, I tried to share a blank document. It won’t do it, interesting. Word is open but blank. Okay, so I already wrote it and only afterward did the sharing.

There’s a relation, say, to—or actually this is often marked as phi, predicate. Doesn’t matter. Between x and y. Say x is the father of y. So R is the relation “to be the father of,” and it holds between x and y. Okay? Or the relation R could be “to be taller than.” Then I say Reuven is taller than Shimon, so x is Reuven, y is Shimon. Okay? So this is just the formalization of some relational statement. This is called a two-place predicate, meaning a predicate that talks about a connection between, or a property of, two objects, or a relation between two objects. So when I, when I put x in place of y here, then this is actually self-reference. Because the relation between x and something else is applied when the something else is x itself. Usually relations are between x and y, between two objects. When the second object is myself, or the referring object itself, that is what’s called self-reference. I think that’s the best possible definition of this matter.

Maybe an example, an example of something that already begins to let us smell the difficulty a bit: I often thought to myself whether I am my own brother. Seemingly yes, because we have the same parents. I and myself have the same parents. Two people who have the same parents are brothers. You understand that this is what—say when I say R of x, the relation R between x and y is “to be brothers.” x is the brother of y. Okay? Now I put x in place of y, and then I ask whether x is the brother of x. So if the definition of brother is to have the same parents, seemingly—and that is seemingly the definition of y, so that is seemingly the definition of R, of the relation—then it’s true. x and x have the same parents, of course. And that basically says that a person is his own brother. But x and x are the same x. What do you mean? x is the brother of x, that x is one, not two. Right, it’s one. I’m my own brother. So what does “same parents” mean? One person has one set of parents. Right. Look at Miki Abraham, that’s x, okay? y is the person who moved from Petah Tikva to Lod and works at the Institute for Advanced Torah Studies. That’s y. Now I claim that those two people have the same parents. Of course they’re both me. These are two different descriptions or two different references to me. Okay? x has those parents, y has those parents, and they’re the same parents. So that means x is the brother of y. Of course here I’m choosing different descriptions for the same person. Formally there’s nothing defective here. That is, it meets the criteria. This relation is what’s called reflexive. That is, the thing stands in that relation also to itself. But here you can already feel that there is some sort of problem. That is, to apply relations that in principle are defined between two different objects, and apply them when the second object is the first object itself, can sometimes create certain problems. Paradoxes, we’ll see in a moment, but it can create problems, and it’s not entirely clear to what extent it is really logically permitted, or logically kosher, to do such a thing. There’s self-reference.

What practical difference do these definitions make? It’s the same one, same everything. What practical difference does it make? A practical difference for betrothing a woman. Is there really a practical difference? If I say I betrothed you as the one who moved from Lod to—what was it?—or the one who was willing to be from some company? No, “a practical difference for betrothing a woman” is the yeshiva joke. What “a practical difference for betrothing a woman” means is: if I betroth a woman on condition that I am my own brother, the question is whether she is betrothed or not. If I am my own brother then she is betrothed, if I am not my own brother then she is not betrothed—there’s your practical difference. We’re in the middle of a joke; that’s just a way of saying I’m discussing a topic right now, so leave me alone about practical differences. The practical differences aren’t what interest me; afterward we can discuss what the practical differences are, maybe that will sharpen things more, but the discussion has some significance even without practical differences. I want to discuss whether it is correct to say that I am my own brother, without practical differences. All right?

This matter of self-reference appears in two main contexts. There is self-reference of claims—a claim that refers among other things to itself, or only to itself—and there is self-reference of a person. Some person refers to himself, thinks about himself, perceives himself, talks about himself, loves himself; “a person dwells within himself.” Actions that a person usually does with respect to something or someone else, or relations that a person usually has with something or someone else, and I’m discussing what happens when this is applied to himself. In other words, it comes up in the context of claims and in the context of human beings. Why specifically human beings? Because with other things it’s not entirely clear to what extent they really refer to anything. A stone doesn’t refer to something else, so there’s no point in discussing what happens when the stone refers to itself. Creatures that generally refer to something are human beings, so I use human beings, but if you like, any conscious creature. The point is that there is self-reference of claims and self-reference of human beings, and I want to discuss each separately and then also see common elements, both in Jewish law and in general.

I’ll start with self-reference of claims. I mentioned that the liar paradox is a clear and famous example of self-reference. What is the liar paradox? The origin of the liar paradox is in the New Testament, where someone appears who is a resident of Crete and says the sentence: “All Cretans are liars.” He’s slandering his own townspeople. I don’t know who first thought of it, but at some point someone noticed that there is actually some kind of loop here. Because if that person is himself a resident of Crete, and he says that all Cretans are liars, then he too, as a resident of Crete, is a liar. But if he is a liar, then the sentence is false, which means it is not true that all Cretans are liars, so he is speaking the truth. But if he is speaking the truth, then what he said—that all Cretans are liars—is true, meaning all Cretans are liars; but if all Cretans are liars, then he is a liar, and so on. There is a loop here. That loop stems from self-reference, because the problem here—what causes the loop to repeat over and over—is that when the person speaks about a group of people, he himself is one of that group. There is self-reference here. If he were speaking about a different group of people that did not include him, no problem would arise at all. If he had said, “All the residents of Crete except me are liars,” there would be no problem with that sentence. The moment he includes himself, there is self-reference here, and that creates some kind of loop.

I already spoke once—I don’t remember in what context—just for accuracy, the original sentence in the New Testament is not paradoxical. A sentence like that is actually not paradoxical. I don’t know, many people quote this as a paradox, but there is no paradox in such a sentence. When I say “All Cretans are liars,” and I myself am, say, a resident of Crete, then that means I too am a liar. If I’m a liar, what does that mean? It means that this sentence is false. If this sentence is false, that means it is not true that all Cretans are liars. Right, because the sentence says all Cretans are liars; if it is false, then not all Cretans are liars. Or in other words, there is at least one Cretan who is not a liar. But that one doesn’t have to be me. It could be my cousin, my neighbor from across the street. Then the loop stops. I really am a liar, and therefore the sentence “All Cretans are liars” is false. Why is it false? Because the neighbor across the street is not a liar. That’s all; now you can’t continue the loop. Because I myself remain a liar, and things remain as I assumed before. Everything stops.

Why does it really stop? Where is the confusion that causes people to think that such a sentence is paradoxical? People don’t notice that the negation of the sentence “All Cretans are liars” is not the sentence “All Cretans are truth-tellers” or “are not liars.” Rather, the negation is “It is not true that all Cretans are liars.” Not all Cretans are not liars—you put the “not” in the middle of the sentence. The negation must negate the whole sentence. It is not true that what? That all Cretans are liars. When you say it is not true that all Cretans are liars, you did not say that everyone tells the truth. You said that there is at least one who tells the truth. Maybe everyone, maybe not, but at least one. And therefore, once you understand that this is the negation, that this is the correct way to negate such a sentence, the loop stops; it doesn’t continue.

And Maimonides already talks about this in Boethius’s square of opposition. It appears in Maimonides’ Words of Logic. And there he discusses the question of how we, what the relations are between different types of sentences. A universal affirmative sentence, a sentence built like “Every X is Y.” That is a universal affirmative sentence. A universal negative sentence is “Every X is not Y.” That is universal negative. A particular affirmative sentence is “There exists an X that is Y.” And a particular negative sentence is “There exists an X that is not Y.” Now the negation of a universal affirmative sentence is a particular negative sentence. “All Cretans are liars” means—well, not the negation but the equivalence—no, the negation, sorry. “All Cretans are liars”: when I negate that, what does it mean? I say that there is one resident of Crete who is not a liar. There exists an X that is not Y. The negation of a universal affirmative sentence is a particular negative sentence. So the negation of “All Cretans are liars” means there is at least one Cretan who is not a liar. But that one doesn’t have to be me. Therefore it stops.

How can I nevertheless create a paradox that is a real paradox on this basis? What do you say about the sentence: “I”—not all Cretans—“I am a liar”? Does that create a paradox? Here there is no generalization anymore. I’m not talking about all Cretans; I’m talking about one particular person, about myself. So that claim too does not lead to a paradoxical loop. Why? Because the negation—when I say “I am a liar,” that means this sentence too is false. When this sentence is false, what does that mean? It means I am not a liar. But what is a liar? A liar means someone who always lies. I am not a pathological liar. There are some statements of mine that are true statements. There are some statements that I say that are true statements. Not all of them are like that. Or in other words, when I say “I am a liar,” here too there is some hidden generalization. Not a generalization over people like “All Cretans are liars”; when I say “I am a liar,” it means all my statements are false. There is a hidden generalization here. And therefore when I negate it, it means there is one statement of mine that is not false. But that is not necessarily the statement I just said now. And again the loop stops.

To create a loop that one cannot escape, meaning an infinite loop that must continue, I need to write, I need to refer to a specific sentence with no generalization whatsoever. When you say: “Sentence A: Sentence A is false.” Meaning the sentence talks about itself. Here there is no generalization at all; this is a sentence that talks only about itself. Not about a group that includes it, not a group of people and not a group of statements, but a claim that talks only about itself. That is pure self-reference, because the previous cases of self-reference were reference to a group that I am part of, so there was also an element of self-reference there, but not pure—it wasn’t only self-reference. This sentence is a case of pure distilled self-reference. The claim refers only to itself, not also to itself but only to itself. And here there really is a paradox with no way out; you can’t stop this loop. If the sentence is false, then that means it is true; if it is true, then it is false, and so on. That is the proper paradox.

So this analysis can demonstrate to you why there is some connection that everyone senses between self-reference and paradoxicality. When you hide the self-reference or wrap it in a broader group, it may be possible to escape the paradoxicality. But when there is pure self-reference, reference only to myself, that is already prone to circularity and paradoxicality. Additional paradoxes. What does he say about that? You hear? What’s the example? So if I say only “I am a liar,” is that paradoxical? Is there a loop? No. When you say “I am a liar,” you’re saying all my statements are false, so there is still a “all” there. But if I say “Sentence A: what does sentence A say? Sentence A says sentence A is false.” That is reference to only one sentence, not to a group of sentences. So that is pure self-reference, and here indeed there is a paradox with no way out. Okay? Yes, yes.

Now there is a whole series of paradoxes that are based on self-reference. For example, the barber of Seville. Right? The barber of Seville shaved all the people who do not shave themselves. Someone who shaves himself doesn’t need him, right? So he shaved all the people who do not shave themselves. And now I ask whether this barber shaves himself or not. So let’s see: if he shaves himself, then he belongs to that group of people whom he does not shave, because after all he does not shave people who shave themselves; he shaves only people who do not shave themselves. So if he himself shaves himself, then he belongs to the group of people whom he does not shave, so he does not shave himself. And if he does not shave himself, then he belongs precisely to the group of people whom he does shave, and then he does shave himself. In short, this too is a loop: if he shaves himself then he does not shave himself, if he does not shave himself then he does shave himself, and this is a loop with its tail in its mouth. And again there is self-reference here, because this barber shaves a group of people, and when I refer to him himself, I ask whether this barber shaves himself, a paradox arises.

But notice that the paradox is not created by the mere statement that this barber shaves himself; that could certainly be. A person who shaves himself—there is nothing paradoxical about that; it’s a description of reality. The paradox does not arise from self-reference alone; it arises from self-reference when in the background there is another claim, another assumption, that this barber shaves all the people who do not shave themselves, and only people who do not shave themselves. Okay? So after I assume that assumption, if I add, or if now I introduce the self-reference and ask whether this barber shaves himself or not, I enter a loop. Why am I pointing that out? Because here you can see that even pure self-reference—when I ask whether the barber shaves only himself—even that does not necessarily lead to paradox. It leads to paradox only because in the background I assumed something about the policy of this barber, that he shaves only the people who do not shave themselves, and now I enter a loop. In other words, I already want to hint here that it is not correct to think that every self-reference is paradoxical, even pure self-reference, yes, even reference to just one thing. But it is prone to generate paradoxes, to generate paradoxical situations.

Yes, another example: there is also a paradox, I don’t know who thought of it, but it’s formulated by Anaxagoras I think, I no longer remember who. Some Greek philosopher; he used to teach law. I don’t know why. In any case, they said he taught a student, took on a student and made a contract with him. “If you win your first trial, you will pay me tuition. If you lose your first trial, you are exempt from paying me tuition—apparently I didn’t teach you well enough.” Fine, the student studies with him, finishes the studies, takes the exam and goes off happily on his way. A day passes, two days, a week, a month, a year—he doesn’t pay tuition. So Anaxagoras sues him, of course, the student, that he should pay tuition. What is the judge supposed to do? If the judge rules in favor of the student that he is not required to pay tuition, then the student won his first trial, but then he is indeed required to pay tuition. If the judge rules that he is not required to pay tuition—that he is required, sorry—then he lost his first trial, but if so then he is not required to pay tuition. In short, here too there is some kind of loop, and this loop is created because when you talk about the first trial in which you will lose or win, it could be a trial that deals with this contract itself. There is self-reference here, though a bit more indirect.

There is also, for example, another paradox in set theory. There are sets that contain themselves as an element. Again, there are two kinds of relation between sets or between things in set theory. There is a set that is a subset of another set; say, the even numbers are a subset of the integers. Okay, so the even numbers are contained in the set of integers. But the number two is an element of the set of integers. The number two itself is not a set; it is a number. It is not a set contained in the set of integers but an element of the set of integers. It sounds similar and it is not the same thing.

Now I want to define a certain kind of sets. There are sets that contain themselves as an element and there are sets that do not contain themselves as an element. For example, the set of all sets. The set of all sets is itself also a set. Which means that it itself is an element of itself, right? There’s no principled problem with that, right? It is an element of itself. Meaning there are sets that are an element of themselves. The set of all things that can be described in words—well, that set too can be described in words, so it is an element of itself. And one can suggest several examples. There are sets that do not contain themselves as an element. For example, the set of even numbers is itself not an element of the set of even numbers. The elements of the set of even numbers are the even numbers. That set is not an even number, so it is not an element of itself, right? Or the set of tables. The set of tables is not an element of itself because it itself is not a table; it is the set of tables. The elements of that set are tables. Okay, so it is not an element of itself. Most sets are not an element of themselves, but some are.

Let’s look at all the sets that do not contain themselves as an element. Let’s collect all of them and form a set out of them. A set of sets such that what characterizes those sets is: all the sets that do not contain themselves as an element. And now I ask whether this set contains itself as an element or not. If it contains itself as an element, then it is not included in itself, because I am talking only about sets that do not contain themselves as an element. So it does not contain itself as an element. If it does contain itself as an element, then it does not contain itself as an element. Does it not contain itself as an element? Then it is one of the sets that are included in it, in itself. So therefore it does contain itself as an element. Again the same problem, and again it is because of the self-reference of this set, or the definition of this set, which refers also to itself. So there are loops or paradoxes that arise from self-reference.

Just so we don’t get confused, I want to qualify here the connection, or the affinity, between self-reference and paradoxicality. I’ll qualify it in both directions. There are paradoxes unrelated to self-reference, and there are self-references that contain no paradoxicality. Therefore it is not correct to identify self-reference with paradoxicality. There is an affinity. Something that has self-reference is suspicious; it is prone to paradoxicality or circularity, but it doesn’t have to be such. And vice versa: there are paradoxes that are indeed paradoxes but are not connected to self-reference.

For example, the paradox of the Swedish army. The paradox of the Swedish army goes like this: a commander comes to his class, his unit, whatever, and says to them, “During the coming week there will be a surprise drill. I’m going to suddenly call you up for a surprise drill.” Okay? Now suppose the drill lasts twenty-four hours just for the sake of the discussion, doesn’t matter at the moment. So, a surprise drill lasting a day. There will be a surprise drill of one day. Now people start doing the calculation. This drill cannot be held next Sabbath. Why? Because if it is held next Sabbath, then on Friday evening we already know there will be a drill—it won’t be a surprise. Because it wasn’t on the first six days, and after all it is supposed to happen this week, so clearly it will be on the seventh day. So it is not a surprise drill; we will know about it in advance. So it cannot be on Sabbath.

Fine, but if that’s so, then it also cannot be on Friday. Because when I’m on Thursday evening I do the calculation for myself and say: on Sabbath it can’t be, because it wouldn’t be a surprise drill; but then it must be—and after all it hasn’t happened until now—so that means it will be tomorrow, on Friday. And again that doesn’t surprise me. So it’s not a surprise drill. Now you understand that it also can’t be on Thursday and not on Wednesday—it can’t be on any day of the week. Now here, this is a paradox, and this paradox on the face of it is not related to self-reference. There is no problem of self-reference here. Nothing here refers to itself. It’s also not a circular paradox. There is no “if it’s true then it’s false; if it’s false then it’s true.” It is some kind of paradox not connected to self-reference, but it is still a paradox. It is a paradox because we know that it is possible to conduct a surprise drill. That is, it is possible to surprise us. And it is not true that there is no such thing as surprise drills. So that was an example showing that there are paradoxes not connected to self-reference.

There is, for example, another paradox that indirectly is probably connected to self-reference, but it is a bit tricky. For instance, think about the following paradox. Let us look at the smallest number that can be described in fewer than a thousand letters. Okay? Take any number—you can describe it in various ways. Say the number four: I can say if you add one to itself four times you’ll get this number. That is a description of the number four. Or eight divided by two is the number four. Or, doesn’t matter right now, all sorts of descriptions. In words. I describe the number in words. Or “the second positive even number,” okay? That is also the number four. There are various descriptions of the number four. And those descriptions can be given in fewer than a thousand letters, right? All the descriptions I just gave are sentences with fewer than a thousand letters. So the number four is a number that can be described in fewer than a thousand letters.

Can every number be described in fewer than a thousand letters? Obviously not. How do I know? Because say in Hebrew there are twenty-two letters. A thousand letters gives me twenty-two to the thousandth different combinations. Now some combinations already describe the same number, but even if every such combination described a different number, there is still only a finite number of combinations here, whereas there is an infinite number of natural numbers—I’m talking now about naturals, or integers, doesn’t matter. Since the natural numbers are infinite, a finite number of combinations cannot describe all of them. Therefore there are clearly numbers that cannot be described in fewer than a thousand letters. Okay?

Now let us take the whole set, all those numbers that cannot be described in fewer than a thousand letters. I want to talk about the smallest among them. In value, the smallest—not the shortest description, but take all those numbers that have no description in fewer than a thousand letters, find the smallest one in value, the smallest number of all, and write it on the side. But that number I can describe in fewer than a thousand letters: “the smallest number that cannot be described in fewer than a thousand letters.” What I just said is a sentence containing fewer than a thousand letters. So I described this thing in fewer than a thousand letters, but by definition it is a number that cannot be described in fewer than a thousand letters. Now here too there is basically a paradox, and this paradox is circular: if it can be described in fewer than a thousand letters, then it cannot; if it cannot, then it can. And at least not directly, there is no self-reference in it. Indirectly there is, but not directly. And this is perhaps another example, I said it’s more tricky and requires a more detailed discussion, but in principle it is another example of a paradox not necessarily based on self-reference.

What is an example of self-reference that did not lead to paradox? Yes, the opposite counterexamples. I’ll give you an example: take the sentence “All sentences are made of words.” That is a sentence; it is itself a sentence. “All sentences are made of words.” It’s a sentence; it is itself a sentence, okay? This sentence is made of words, right? Meaning this sentence can also refer to itself, a true sentence: all sentences are made of words, including this sentence itself. Is there any sort of paradox here? Not at all. No problem at all. It is a true sentence, and it correctly describes itself as well. There is no paradoxical loop here, no logical problem arises, there is no problem here; there is clear self-reference here but no paradox is created.

Moreover, take pure self-reference: a sentence that says, “This sentence is made of words.” Here we already have pure self-reference only to this sentence itself, not to a collection of sentences. But this too creates no paradox. Right, it is a sentence that refers to itself and says something true: “This sentence is made of words.” Right, this sentence really is made of words. Okay, so here are examples of self-reference that do not lead to paradoxes.

If you want, I’ll give one more example here that maybe I’ll come back to later, just for our example. Remember the liar paradox? Right, that’s the liar paradox, correct? What’s written here. Let’s call it this: “Sentence A is false.” Fine? What do you say about the following sentence? On its own it can stand. What? On its own it can stand. We’ll talk about “on its own”; these are two unrelated sentences, they’re independent. Okay, that can also be perfectly fine. It could be. This is not a paradoxical sentence, right? No, no. Sentence A is paradoxical, because if it is true then it is false, if it is false then it is true—that is the loop we talked about. This sentence, there is no problem with it: if it is true then it really is true and everything is fine. Right? Yes. But that’s not exact, and that’s why in a certain article I once called this—if that is a paradox, this thing is an anti-paradox. Why? Think what an ordinary sentence is, an anchored sentence, a kosher sentence. When I say “The wall in front of me is white.” Okay? Then that sentence is either true or false, depending on what color the wall is. If the color of the wall is white, then the sentence is true; if the color is something else, then this sentence is false. Meaning, an ordinary sentence is a sentence that can be either true or false, but only one of the two, depending on the state of affairs it describes. Right?

A paradoxical sentence, like the one I described here, is a sentence that can be neither true nor false, right? You cannot assign it any truth value. If it is true then it is false; if it is false then it is true. It has no defined truth value. Okay? This sentence has two possible truth values. You can assume it is true, and then it indeed comes out true; that is consistent. But you can also assume it is false, and that too comes out consistent, because if it is false then its claim that it is true is indeed false, which means it is false. And then that means that this sentence can be both true and false. So it is not a standard ordinary sentence, it is not the normal kosher sentence we are familiar with. A sentence we are familiar with is either true or false; it cannot be both. This sentence can be both. Therefore I call this—it is not a paradox and not a regular sentence—I call it an anti-paradox. An anti-paradox is something that receives two truth values. A regular sentence or regular claim gets one truth value, either true or false, but not both. An anti-paradox gets two truth values. It can be true and it can be false at the same time.

It somewhat contradicts the law of non-contradiction. The law of non-contradiction says that if a certain sentence is true, it cannot be false. Either it is true or it is false. No, this sentence, for example, can be true but can also be false. How? What do you mean, how? How can it be false? If it says “This sentence is true,” in what situation would this sentence be false? Assuming that this sentence is—let’s assume it is false. And now let’s check its content. What is its content? Its content says that it itself is true, but that is false according to our assumption, meaning that it really is false. So that is consistent. The assumption that it is false is consistent, and the assumption that it is true is consistent. I understand. Therefore it is impossible to decide whether this sentence is true or whether this sentence is false. It can be this and it can be that. And both, in fact. Okay? That is called an anti-paradox.

But notice once again, there is a sentence here that refers to itself in this sense; there is self-reference here just as there is self-reference here. Only here the self-reference creates a paradox, and here the self-reference does not create a paradox. It creates what I called an anti-paradox, but not a paradox. So this is another example of self-reference that does not create a paradox.

If so, basically our conclusion so far is that although self-reference is prone to creating paradoxes, the connection between self-reference and paradoxicality is far from necessary in either direction. Not every self-reference is paradoxical, and not every paradox stems from self-reference. This relation does not hold in either direction. Okay.

Now I come to type theory. Bertrand Russell was one of the greatest philosophers and also mathematicians of the twentieth century, and he wrote a monumental book with Whitehead, three thick volumes where it’s all just formulas. To get through that is a project. A real life project. They wrote this book; it’s called Principia Mathematica, incidentally the same name as Newton’s book, Principia Mathematica. And in the introduction—the introduction I did get through—in the introduction to that book, at the beginning of the first volume, he talks about type theory. There he proposes some kind of solution to paradoxes of self-reference. And he proposes the following solution.

Basically, he says—without going into details—he constructs some hierarchy of statements in language. There is a hierarchy of statements; every statement belongs to a certain type. Type A, type B, type C, type D. Every statement is in some type. He proposes that hierarchy there, doesn’t matter right now; he builds a hierarchy of statements. And the rule he proposes, in the grammar of the language he proposes, is that a statement cannot refer to statements that are in its own type, only to statements lower than it, in lower types. And once we adopt that rule, then in fact self-references do not appear at all, and of course all the paradoxes are avoided as well. Because a statement that refers to itself is doing something illegal. It refers to itself, and it itself belongs to the type that statements in it cannot be referred to by that statement, according to the rule that a statement may refer only to statements lower than it in the hierarchy. Okay? Remember that I am my own brother? Meaning that statement is in the same type as itself. Therefore it cannot refer to itself, because it cannot refer to statements from that type. Okay?

So that is how, very briefly, he proposes to solve the problem of self-reference, of paradoxes. Now I, when I read this, and later also when I read about it and people refer to it as one of the proposals to solve paradoxes—because in one of the comments on the previous column where I wrote about it, the last column where I wrote about it, someone noted that Bertrand Russell there does not really present this as a solution to paradoxes. He presents it as a precise language that will not contain paradoxes, and that is not the same thing. Mathematicians can define for themselves a language such that using it prevents paradoxes from appearing, for the convenience of their work. Okay? That does not mean they solved the paradoxes. It means that mathematics is conducted in a language in which paradoxes cannot appear. It is impossible to represent paradoxes in that language. That does not mean you solved the paradox; it means that this paradox will not bother you when you speak in that language or engage in mathematics. Okay?

But after Russell, many others—philosophers and others—did see his proposal as one of the possibilities for solving paradoxes of self-reference. If one adopts that rule, then one basically solves the paradoxes of self-reference. On this matter I am not willing to accept that proposal as a solution. It is not a solution to paradoxes. Why not? Basically as I described earlier, because it’s like saying: let’s translate the paradox from Hebrew into English, and in English we won’t allow self-reference. Fine. Did I solve the paradox? No. I’m just using a language in which it cannot be expressed. So that means I evade the paradox or forbid expressing it, but I don’t solve it. The paradox exists, except that in your language you don’t know how to express or describe it. That is a problem with your language; it is not a solution to the paradox.

In a borrowed sense, my feeling is that such a solution to paradoxes is more or less Stalin’s method of solving problems. Stalin solved all the problems in the Soviet Union that way. If someone raised a problem, he cut off his head. Then that was it—you’re not allowed to raise problems, so they don’t exist. In my eyes, the solution of type theory is a very similar solution, of course less violent, but very similar. It merely forbids expressing the problems; it does not solve them. You build a language in which it will be impossible to express the problems—what have you gained? The problems exist. You haven’t solved them.

So therefore my feeling is that this solution, this so-called solution, of type theory does not really solve the paradoxes of self-reference but only forbids expressing them. But more than that, this solution of type theory is problematic because it throws out of the language, or forbids expressing in the language, many many sentences or claims that are not paradoxical, that are perfectly kosher. Let me remind you of the examples I gave before: “This sentence is made of words,” or “Every sentence is made of words.” A completely kosher sentence; it contains self-reference but is not paradoxical, raises no problem at all, it is even a true sentence. In Bertrand Russell’s language it will be impossible to say that sentence, because it is a sentence that refers to itself and also to other claims belonging to its type, and that is illegal in that language.

Now what justification is there for throwing perfectly kosher sentences out of the language just like that? There is no justification for it. If I formulate another angle of the same problem, actually these two problems are connected, I could have solved self-reference much more simply. Every sentence that creates a paradox is illegal. Thus I also solved the paradox of the Swedish barber, not just paradoxes of self-reference, right? It’s a much more elegant solution than Bertrand Russell’s solution. Because here I throw out only the problematic sentences and not all self-reference, and that’s it, and I’m left with a language in which there are no paradoxes. I solved the problems of the universe.

So why does Bertrand Russell take a more radical or more violent solution? He throws out of the language many more sentences for which there is no reason to throw them out. Throw out only the paradoxes—they’re what cause your problems—throw them out! The point is, if we formulate the problem a third way, that there does not seem to be any real justification for the rule Bertrand Russell proposes. What justification is there for not allowing self-reference in a language? After all, things can in principle refer to themselves, so what justification is there for building a language that does not allow self-reference? You can say the justification is that it prevents paradoxes. That means it is useful to you, but I am asking why it is true, not why it is useful. Or after all, to prevent paradoxes I have other proposals too, so they have the same justification. So which proposal is the correct one? They all have the same justification. We see that this justification cannot really serve as a justification for Bertrand Russell’s proposal.

I can propose a hundred thousand other proposals that would solve the… It is forbidden to express in the language sentences of more than two words. Fine, or one word. That too solves all paradoxes; I don’t know of any paradox that can be expressed in one word. So what? One can propose a million languages in which paradoxes cannot be expressed. That is not a solution to the paradox, unless you show me that in a genuinely logical and independent sense this rule can be explained—the rule that a claim cannot refer to its own type or above, only to lower types. If you convince me that this is true, and then I adopt it, and the result is that indeed there will be no paradoxes of self-reference, excellent. But the fact that it prevents paradoxes of self-reference cannot by itself serve as justification for the rule. With that kind of justification I could do what I said before: throw out every sentence that is paradoxical; I have justification because it is paradoxical.

This reminds me, yes, of another form—once I took a course at Tel Aviv University on paradoxes, taught by Anat Biletzki; she also wrote a little book in the University on Air series about it. So she also brought there another type of proposal for solving paradoxes, and that is basically to talk about three-valued logic. Three-valued logic basically means that binary logic, which is our usual logic, treats every claim in one of two ways: either it is true or it is false. It must be either true or false, certainly not both and not some third thing either. Okay? That is binary logic.

Ternary logic, three-valued logic, is a logic that has three truth values for each claim. Not only truth and falsehood, the two classical truth values, but true and false, T and F, yes, true and false, T, F, and P, paradox. Meaning every claim can be either true, or false, or paradoxical. And now there is no problem: you have a paradox—what is the paradox? You cannot assign it a truth value, because if it is true then it is false, and if it is false then it is true. No, you can assign it a truth value. Assign it the truth value P. There are three truth values, not only true or false. True and false you can’t assign it, but there is P—assign it P, so you can assign it a truth value and everything is fine.

This solution smells very much like the previous ones. Yes, of course this solves nothing at all. What? That is what I call stretching a question mark into an exclamation mark. You have a question mark—stretch it and turn it into an exclamation mark. Yes, that does not answer the question; it turns the question into an answer. It’s not an answer, not a response, not a solution. The fact that you assign such a sentence a truth value that you call P—what exactly have you done? In the end you merely gave it a name. After all, what bothers me is that sentences are supposed to be either true or false; there is no third possibility. So when I suddenly find a third possibility and you give it a name, that does not justify the possibility, it does not justify the existence of that possibility, it doesn’t explain it. You merely gave it a name.

These are the kinds of solutions that try to solve real problems by changing a definition. A definition of language, or a definition of the number of truth values in your logic, and basically they see the definition as the solution to the problem. But definitions do not solve problems. A definition is a definition. You can do—depends what use you make of it—but a definition, changing a definition, does not solve a problem. Changing a definition merely helps express the problem, exactly as in type theory. Therefore all these kinds of solutions are not really solutions.

And again I say, I accept that reservation in the comment I mentioned earlier. That does not mean Bertrand Russell’s proposal is worthless. It may be that Bertrand Russell says: you’re right, it is not a solution to paradoxes; I did not come to solve paradoxes. I merely came to propose a conceptual framework or language within which mathematical work can be conducted without paradoxes popping up. Without bothering me. I leave them outside. I haven’t solved them, but they won’t interfere with my mathematical work. So if I stick to that language, then that will happen, meaning the paradoxes won’t get mixed in there and won’t disturb me, won’t bother me. And that may be useful for mathematical work. That may indeed be useful. But as a solution to paradoxes it certainly is not a solution. With God’s help. What? That’s what’s called “with God’s help.” You give it another name, give it a framework, and then we’ve solved the problem. Exactly.

Okay, so what I did so far was to try to show self-reference of claims. And claims that refer to themselves or to a group that includes themselves—that is self-reference of claims. Even the paradox of the barber of Seville, yes, is basically a paradox of claims. Because the claim is that the barber of Seville shaves all the people who do not shave themselves. And then the question is whether he shaves himself—that’s another claim—or whether he does not shave himself—that’s another claim—and the paradox is created by the relation between those claims. In the end it is a logical paradox, a paradox of claims.

When I speak about self-reference of human beings, that is something else, not necessarily related to paradoxes at all. For example, when I say that, that—say I’m speaking about, yes, I spoke about this in the course on philosophy. When I think in logic, when I investigate the way human beings think, investigate the mode of operation of the intellect—here I mean the intellect and not the brain—what tools do I use? I obviously use my tools of thought, right? When I do logical analysis, I do it by means of my own tools of thought. But my tools of thought are the objects of the research. So there is self-reference here. This is not self-reference in the sense of the paradoxes, because someone could come and say: I don’t see any paradox in this. Right, I use the tools of thought to investigate the mode of operation of the tools of thought themselves. What is the problem with that? Why not? Why can’t one do that? There are people who might feel some discomfort here and say that one cannot do this, but think about it—it is not trivial that one cannot do this. On the face of it, what is the problem? Right, the intellect thinks about itself and investigates itself, and that’s all.

So here there is self-reference that is not on the logical plane, but on the human plane. I refer to myself. It is not a matter of a claim referring to itself and a loop being created in terms of its truth values. There is no problem of truth values here and none of that; there is simply a person’s relation to himself. Okay? Rabbi, this is limited, because I refer with a tool—it is something limited. I cannot break through to look at it from the outside, because I look at it only from the inside. Okay, so the point is as follows. Someone might come and say: then apparently your chance of succeeding or arriving at a full description of the way thinking works is small; it doesn’t exist. But that does not mean the conclusions I did reach are not correct. Not necessarily, in any case; I do not see a logical necessity to say that.

Again, there is something disturbing when we suddenly understand that the intellect here is investigating itself, but on the simple logical level, on the face of it, that should not necessarily be problematic. Right, the intellect thinks about itself and investigates itself, and that’s it. It may miss things, but there is no principled obstacle to doing it, to using the intellect to investigate the intellect.

Look, this brings me to the poetic part of the evening. I came to the conclusion, after I wrote two columns—two columns and two turtle doves, as they say—with a gap of several years between them. Column 81 and column 365. Both columns dealt with songs by Shalom Hanoch. One of them deals with the song “A person dwells within himself.” I assume it is familiar. And the second is “For my song is an echo in the wind.” Know it? “For my song is an echo in the wind, from the open me,” and so on. Yes, that is the second song. And suddenly I noticed, with no connection at all—that is, I decided to write about both because both stirred thoughts in me—and suddenly I saw that it is the same thing, the same kind of thought.

So look, for example, let’s start with “A person dwells within himself.” Okay. Look here: “A person dwells within himself, yes, a person dwells within himself, sometimes sad and bitter, sometimes he sings, sometimes he opens a door to welcome an acquaintance, but mostly a person closes himself within himself.” And then “A person dwells within himself,” etc., “in some stormy city or in some village, a storm passes, his house breaks.” His house is of course himself, because he dwells within himself, “but mostly a person is a stranger even to himself.” So “a person dwells within himself,” but “mostly a person is a stranger even to himself.” Yes, this matter is very interesting to me.

A little later he also brings a spouse into this matter, whether she is part of him or not, yes, “his wife is as his own body.” In any case, fine, we won’t read the whole song. But what lies behind this? And again, don’t take this like Kurzweil and Agnon, okay? I’m not committing myself to the claim that Shalom Hanoch thought exactly these things when he wrote the song, but I do tend to think that it was there in some form, perhaps not consciously or only half-consciously, but that intuition was there behind the lines when he wrote this song.

When I say “A person dwells within himself,” there is of course clear self-reference here. This is self-reference. A person lives in a house, a person lives in a cave, a person lives in a tent, a person lives in a village, a person lives in something. So this is a two-place relation, R of x and y, right? Now I put x itself in place of y. A person dwells in—the y—is himself, x. R x x. This is self-reference, right? Now when a person refers to himself, what does it mean to open a door? To open a door means to look outside. But the door is part of what he himself is; it is not something that wraps around him. So what does it mean that he opens a door and meets the outside, to welcome an acquaintance? Yes. But mostly a person closes himself within himself. Because if the house surrounding you is yourself, then it is doubtful to what extent you can get out of it at all, yes? This is postmodernism. You are trapped within your own narrative.

And it is especially beautiful: “A person is a stranger even to himself,” yes, the last line of the second stanza. “A person is a stranger even to himself” basically means that I am still treating this two-place predicate as a predicate with different x and y. Why? When I say “A person dwells within himself,” my reference to the person and to the house in which he dwells, which is also himself, is as though there were two different objects here. I am a stranger to myself. When I function as my own house, true, ostensibly it is I and myself, but two strangers who happen to coincide.

Think again about what I said at the beginning, that I am my own brother because we both have the same parents. Okay, so if I refer to myself in two different ways—Miki Abraham or, yes, the one who teaches at the Institute for Advanced Torah Studies—two descriptions that happen to coincide and describe the same person, but two different descriptions: Mr. So-and-so and so-and-so have the same parents. If I had said it that way, people wouldn’t feel any problem or strangeness. They would understand that these are brothers. And then I would tell them no, no, it’s the same person himself. Fine, okay, a description is a description.

So what creates the problem, basically? What creates the problem is that on the one hand reference is always between x and y. But on the other hand, when I refer to myself, the y to which I refer is x. The one who refers is also the thing referred to. So a person who refers to himself actually sees himself as a stranger. And in fact self-reference forces you to perceive yourself as a stranger, because otherwise what are you looking at? You are the one looking, not the thing being looked at. If you are the thing being looked at, then basically you function as something foreign to you as observer. You as subject, yes, the observer, are foreign to you as object, the one being observed.

Remember the thinking by means of whose tools we investigate our modes of thinking? There too, in fact, the use of those tools to investigate is my subjective use. Here those tools function as my tools. When I examine the rules of logic that I investigate—not the tools I use but the objects of investigation—there, although they are the same tools, the tools I use are me. But when I refer to them, I refer to them as something foreign. I see them as something outside me and investigate them as though they were y. I ignore the fact that in this particular case y is x itself. The moment I can make that break and see myself as something foreign, to look at myself from outside—sometimes one can put it like that, I look at myself from outside—that is basically to say that a person dwells within himself and yet is sometimes a stranger. The ability to look at myself from outside is the ability of estrangement. Defamiliarization, yes? To make into a stranger. Okay? Yes, “I am a stranger and a resident among you,” but it says not with myself, someone noted in the chat. Here it is a stranger and a resident with myself.

So this claim that a person dwells within himself and usually is a stranger to himself—he is a stranger when he dwells within himself. Because if he dwells within himself, that means he refers to himself as something other within which he dwells. So the dweller is me, and the house in which I dwell is something else that just happens to be another description that is also me. So this is the capacity for estrangement, yes, to see myself as something foreign, which I think is a beautiful expression of the problem of self-reference. The question is whether I am really referring to myself or whether I construct some model of myself that is foreign and refer to that. So what difference does it make that it is a model of myself? It is still something else, something foreign.

Just to complete the picture, the second song—yes, “For my song is an echo in the wind.” A beautiful song in my eyes. Yes, “For my song is an echo in the wind, my writing sent forth, the track of my life, my longings, the echo of my prayers. For my song is a leaf in the wind, forgotten and blown away, it is the gentle light opening in my nights, it is you coming toward me.” Yes, “Where is the darkness all around. If only you were listening.” Why that’s in the same line I don’t know. “You come and go toward me,” of course again there is some hint here, but in a moment. “Along my way, her sign,” and so on, okay. Here he continues: “For my song is the gust of wind, my open window, the spring of my strength,” etc., “comes and goes toward me.”

So yes, my wife Dafna once asked me why this song is called “Song Without a Name.” That is the song’s title. Incidentally, it’s a song with a name. Its name is “Song Without a Name.” Or perhaps he means, no, I don’t want to give this song a name, so I call it “Song Without a Name.” Yes, once in the Israel Museum there was hanging there—one of the reasons I don’t go to museums is exactly this reason—there was hanging there a frame, a frame, and inside it an empty canvas, and above it was written: the name of the work is “Wooden Frame with Metal Hanger.” That is the name of the work. Nothing is painted. Okay? So that’s roughly “Song Without a Name.”

So there can be two interpretations: either he really means I don’t want to give this song a name, or he means no, this is the name of the song, the name of the song is “Song Without a Name.” I tend toward the second interpretation. And I said to Dafna that I think there is a good reason why this song is called “Song Without a Name.” Who is the song the song is talking about? “My song is an echo in the wind, my writing sent forth,” and so on, “the track of my life, my longings,” and the like. What—who is the song? This song, I think, is this song itself. Is it talking about itself? Yes. Does this song refer to itself? Yes. This song itself is an echo in the wind, and the song that describes itself is itself the song around which the descriptions revolve. So it cannot have a name? Right. Because to give something a name you have to be something outside it. “Every person has a name,” as they say, “given him by his father and mother.” Today I’m apparently in a Hebrew-song mode altogether, yes? “Given him by his father and mother,” and so on. “Given him”—they gave it to him. Meaning a name is something given to you. It is something by which the environment refers to you through your name. I do not refer to myself through my name. Now something that refers to itself is something without a name. Because it is not someone else referring to it and giving it a name; it refers to itself.

You see that this is exactly “A person dwells within himself” and usually is a stranger; it’s the same thing, it is self-reference. And in both places we see that self-reference on the one hand—if you really refer to yourself self-referentially, you cannot give it a name, because you have no ability of estrangement. Then you are referring to yourself; you are not referring to some y that just happens to be x, you are referring to x. So it has no name. It is not—that is not reference. Reference has to be between two things. If you really want to refer to yourself, then “a person dwells within himself,” so you have to be a stranger. You have to have the ability to look at yourself from outside, to place yourself at the side, to give yourself a name. You give yourself a name, but the name you give is to some model that represents you, which from your present point of view is something else that is not you.

These are, I think, nice poetic expressions; I think they present the matter well. Again, I don’t know how aware Shalom Hanoch was of it, but it seems to me it was there somewhere. But I think this is a good presentation of the second problem of self-reference—not the logical problem I described before of claims that refer to themselves. There there is no problem of subject and object. Here I am speaking of self-reference as a phenomenon we might call psychological or epistemic, cognitive. I know something—normally to know something is to look at something outside me, and here I look at myself. I myself am the object I am looking at, and I am also the observer of the thing. Okay? So this too is self-reference, but notice it is not necessarily paradoxical in the logical sense. I refer to myself, I look at myself. Fine. It is not paradoxical in the same sense that self-reference of claims leads to a logical paradox, meaning that one cannot assign a truth value there, one can assign neither truth nor falsehood. Here the problem is not that I cannot talk about the truth or falsehood of a claim. We are not talking here about claims; this is a mental state in which a person refers to himself. But it creates various problems, which we will discuss later, that in some way do resemble the self-reference of claims.

Want to say something? Comment? Ask? Rabbi really gave Shalom Hanoch enormous praise. Yes, he really is excellent. Artistically too, in my opinion, he is excellent, but here these really are insights that I think, at least to some extent, he was aware of too; it was there somewhere in his consciousness when he wrote this. If it’s one song and another song, then apparently there is something there. Yes, he probably has some philosophical mode, a philosophical temperament. Okay. Great. All right, Sabbath peace, Sabbath peace, all the best. Sabbath peace, thank you.