Artificial Intelligence – Lesson 12 – Rabbi Michael Abraham

[Rabbi Michael Abraham] Good. So last time we more or less finished the general, philosophical introduction, where we dealt with the question of what thought is, what a human being is. To the extent that it’s possible to define these things, or illustrate them, or clarify them, because “define” is a bit too strong a word. I just want to remind you of a very important point for what follows, one that also stirred up some arguments and took us a bit of time. When we talk about thought, or about a thinking creature, then in my opinion at least, we need to require some kind of exercise of judgment, awareness, and things like that. Without those things, it may be that certain operations are being carried out that syntactically are operations of thought — yes, structurally and formally they are acts of thought — but we still wouldn’t call them thinking. The example I used was water. When we describe the motion of water, we use the Navier-Stokes equations, which are very hard equations to solve. By the way, not long ago I saw something AI did with them; I don’t remember exactly what. And the water, supposedly, solves those equations. In other words, the description of its flow is basically a solution to the equations. And I assume none of us would relate to water as a thinking entity. So that point is going to be very important later on when we move to computers and artificial intelligence, and ask whether what’s happening there is really an act of thought. But that’s just a reminder of what we saw in previous sessions.

I now want to start getting into the technical issues themselves. Obviously I’m not going to give a course in computers here, and I’m not really an expert. But I do want to try to demonstrate what this creature called a computer actually is, because in my view it’s very important to understand that concretely before getting into all sorts of conceptual and philosophical discussions. Otherwise I think those discussions just float in the air. So I’m going to devote today’s lecture, and probably the next one too, to a few technical matters.

So maybe I’ll begin with a very short introduction to logic. The connection between logic and computers is pretty well known, but I want to make it concrete. Classical logic deals with inferences. That is, I have certain premises, and from those premises I derive a conclusion, or conclusions. The tools I use in order to derive the conclusions are logical tools. For example, if I say: all human beings are mortal — that’s one premise. Socrates is a human being — that’s another premise. Conclusion: Socrates is mortal. Okay? So I derived the conclusion from the two premises. There are logical tools that help me describe or express this process of moving from the premises to the conclusion. Okay?

Now those tools assume that the premises and the conclusion are symbolized, formalized. To formalize is to put something into a form. “Formalize” comes from “form.” We’re talking about formalizing, meaning to symbolize or formalize. So we want to formalize propositions in language. And in order to formalize propositions in language, what we basically need to do is begin with a few basic atoms and try to build from them a representation of more complex propositions.

So for example, suppose I want to deal with the proposition: both Socrates is mortal and Yankele is mortal. Okay? Then I’ll have to say that “Socrates is mortal” — let’s denote that by A. “Yankele is mortal” we’ll denote by B. And in order to represent the complex proposition I just described, I also need to use what’s called a logical connective. In this case, the connective “and.” All right? A conjunction. So A and B is the representation of the sentence — the logical representation of the sentence I said before: both Socrates is mortal and Yankele is mortal. So my atoms are “Socrates is mortal” — that’s one atomic claim. “Yankele is mortal” is another atomic claim. I denote the first by A, the second by B. The complex sentence will be denoted as A and B.

Now when I ask how many logical connectives there can be — at first glance, infinitely many, right? As wild as our imagination goes. There’s “and,” for example. There’s “or,” for example. Either Socrates is mortal or Jacob is mortal. There can also be “if Socrates is mortal, then Yankele is mortal” — that is the connective of implication, the if-then. Okay? So there can be all sorts of connectives, and the question is whether I can say something about how many connectives there can be in total. This is very important when we come to deal with formalization, because we want to see that we have a way to represent all sentences in language. Okay? So first of all I want to identify what kinds of sentences can exist at all. And since I build complex sentences using logical connectives, the first question I need to ask is: how many logical connectives are there?

Okay. Now in order to answer that question, let’s look for a moment at a more formal description of the matter. So look. Let’s take, for example — I’ll share a file for a moment. Okay, ignore all the text; that’s text for me. But look at the table. All right? So in the table we basically have two propositions, P and Q. Those are our atoms. And you can see here we have P or Q, right? This symbol is “or,” a logical connective that we mark as “or.” So: P or Q. Okay?

Now I build what’s called a truth table. What does that mean? A truth table basically tells me: given the truth values of the atomic propositions, P and Q, I want to know the truth value of the complex proposition. Okay? Now if both P and Q are true — true and false here are marked as T and F — so if both are true, then what is the value of P or Q? It’s true too, right? Suppose I say to you: either Socrates is mortal or Yankele is mortal. Now the truth is that both of them are mortal. So what do you say about the sentence “either Socrates or Yankele”? Obviously it’s true as well, right? If each of them is true, then the compound sentence is also true. But it will also be true if only one of them is true, right?

For example, look at this row. If Socrates is mortal — that’s P — then P is T, true, yes? And Q, “Yankele is mortal,” is not true, so that’s false. What is the value of P or Q? Still true, right? Because I’m saying either this is true or that is true, so it’s enough that one of them be true for the compound sentence also to be true. Right? Same thing here. If in this case Socrates is not mortal but Yankele is mortal — meaning P is false but Q is true — then what is the truth value of the compound statement P or Q? True. Same thing, right? If one of them is true, then the value of P or Q is also true. The only case in which P or Q is false is the last, fourth case, in which both P and Q are false. Okay? If both P and Q are false, and I say either this is true or that is true, that’s a false statement, right? Because neither of them is true.

Okay, so this is just an example of a particular logical connective, “or,” and you can see that this connective can be represented by a truth table. Okay? In other words, the truth table tells me: given the truth values of the two atomic propositions, I want to know the truth value of the complex proposition. That’s what the table does. Okay? This table will always have four rows, right? Because what combinations can there be? Either both are true, or this one is true and that one false, or this one false and that one true, or both false. There is no other option, right, with P and Q. So there’s no other option for P and Q. Therefore a connective — in this case a two-place connective, because it connects only two atomic propositions — is basically a truth table represented by four rows and three columns. All right? The two columns on the left, these two, are the input columns. It takes P and Q as input, and this is the output column. That’s the output column. This column tells me what the logical connective actually gives, what the result of the logical connective is. Okay? That’s why it’s called output. You can already start seeing why this is related to computers, but we’ll get to that in a moment.

So that’s, for example, the case of P or Q. Now let’s look at a few more connectives. So look at this table. You see? P or Q — I described that above, right? That’s what we saw before. Now here I’m taking — I said that in all these tables there are four rows, right? That’s determined by the two inputs, right? Those are the four possible combinations. Stop me if I’m not clear or if I’m going too fast. For some of you this is definitely too slow, so I’m trying to maneuver here between people who have background and people who don’t. So stop me if I’m wrong or if I’m not doing this properly.

So we have four rows determined by four different possible inputs, and each such column is a connective. So this column here is basically the connective “or.” It tells me what the result is if the compound proposition is P or Q. Let’s look at this. This is the connective “and,” P and Q. You see how the symbol flipped here? You see? This is the symbol for “and.” “And” is what’s called conjunction, and “or” is disjunction — those are the foreign terms, not so important.

Now look at how I built this column. Let’s see. When I say that both P and Q are true, when is this compound sentence true? If P is true and Q is true — take this row — then the result, of course, is that P and Q is also true, right? But in every other case the result will be false. Why? Because when P is true and Q is false, it’s not true to say that both P and Q are true. Right? And when P is false and Q is true, again it’s not true to say that. And when P is false and Q is false, of course it’s not true to say that both P and Q. Okay? So that’s how I built the connective “and.” Okay? That’s this column.

Now let’s move for a moment to this one; it’s easier for me. This is the connective of equivalence. What does that mean? P is equivalent to Q. You see the symbol — those three lines, like an equals sign with an extra line. Okay? It’s not equality, it’s equivalence. “Equivalent” is in logic; “equal” is in mathematics. Okay? So this column, which marks the connective of equivalence — let’s see why it’s built this way. Look: if P and Q are both true, then they are logically equivalent, right? This is true and that is true. If P is true and Q is false, then they are not equivalent, right? Because P is true and Q is false. Same thing if P is false and Q is true — again, you see? The result is false. And if both are false, then the result is true, right? Because they are equivalent. So equivalence gets the value true in the first and fourth rows, but false in the two middle rows. Okay?

[Speaker C] Does that answer the question—

[Rabbi Michael Abraham] If they’re equal?

[Speaker C] I can’t hear. Does that answer the question of whether they’re equal? And that’s what “equivalence” means?

[Rabbi Michael Abraham] Yes. I said: in mathematics it’s called equality; in logic it’s called equivalence. Okay? So say, for example, let’s say the switch is up — let proposition P be “the switch is up.” Proposition Q is “the light is on.” So P is not equal to Q. The sentences have different meanings. Right? “The switch is up” and “the light is on” are two sentences with no connection between them in terms of content. But they’re considered logically equivalent because whenever this is true, that is true, and whenever this is false, that is false. If the switch is up, then the light is on; and if the switch is not up, then the light is not on. Okay? So therefore “the switch is up” and “the light is on” are equivalent statements. They are not equal, they are not identical, yes, their contents are completely different. But logically they are equivalent. Equivalence means: if this is true, that is true. That’s called logical equivalence. Okay? And if this is false, that is false. Okay? So that’s this column.

Now this column I left for the end; the good things are always left for the end. This column is a problematic one. Let’s try to think why. Look, this column basically tells me: if P then Q. You see the implication arrow? It’s called implication — if P then Q. P is the antecedent of the implication and Q is the consequent. Okay? Now how do I fill in this truth table?

So suppose P is true and Q is true. Forget what’s written here, all right? What would you put here? If P is true and Q is true, can I say that P implies Q is also true? I didn’t understand—

[Speaker D] What does “implies Q” mean?

[Rabbi Michael Abraham] “Implies” means that if this is true, then that is true. For example, with the switch: if the switch is on, then the light is on.

[Speaker D] So here it’s true. Of course it’s true.

[Rabbi Michael Abraham] So let’s take an example. Let proposition P be: the President of the State of Israel is Bougie Herzog. Okay? Proposition Q is: now it is night. Okay? Now I say: if Bougie is the President of the State of Israel, then now it is night. Leave logic aside for a moment — what would you say about that sentence? Just nonsense. Right, what connection is there at all? Or in other words, what that means is that the implication statement is exceptional among all the connectives we’ve seen so far. And why? Because it is determined by the content of P and Q. It is not enough to know the form and the truth values; I also have to know the content. If P and Q were “the switch is up” and “the light is on,” then P implies Q really would be true when both are true. Okay? But if it’s “Bougie is President of the State of Israel” and “now it is night,” then you can’t say that “if Bougie is President of the State of Israel, then now it is night” is a true sentence, because there is no connection at all. The “if” does not carry over to the “then,” right?

So it’s not enough for me to know the truth values of P and Q in order to know what the truth value of P implies Q will be. It depends on what P is and what Q is. It depends on the content. Notice that in the last four columns we went through, it did not depend on the content. For example, if I said “both Bougie is President of the State of Israel and now it is night outside,” that sentence is true, right, if both parts are true. I don’t need to appeal to the content of P and Q in order to determine whether the sentence is true. It is determined only by the form or by the truth values. If you know the truth values of the inputs P and Q, then you know the truth value of the output, the result. Here in this column, that’s not true.

[Speaker B] In this column, that’s not true.

[Rabbi Michael Abraham] In this column, it’s not enough to know the truth values of the inputs in order to know the truth value of the output. And the reason is that the truth value of the output depends on the content of the inputs and not only on their truth values. It’s not enough to know that both are true in order to know that there is an implication relation between them. They can both be true and yet have no connection at all, so you can’t speak of implication. Okay? Therefore implication is a very problematic connective.

And what is usually done in logic, in order to turn it into a formal connective that depends only on truth values and not on the content of the propositions, is to adopt the minimal meaning of implication. This is called material implication. So this is basically material implication. What does that mean? It means: what is the minimum I require of an implication statement? The implication statement tells me, at a minimum, the following thing: it cannot happen that P is true and Q is false. Okay? That’s the claim. In other words, it’s true that this is not enough to describe implication in full, but it is the minimal requirement, and it must hold. That is, it cannot be that P is true and Q is false. If P is true, then necessarily Q is also true. All right? That’s just another way of saying the same thing. To say that if P is true then Q is true — what does that mean? It means that if P is true, then it cannot be that Q is false. Right? That’s the same thing.

So this is the minimal interpretation given to implication in order to turn it into a formal relation, a relation of formal logic, that does not go into the question of what P and Q are, but wants to suffice solely with their truth values. Okay? And now I fill it in like this: if both P and Q are true, then indeed we do not have a situation here in which P is true and Q is false. Therefore the result is true according to the minimal requirement I set. The only place where the result will be false is the second row. Because you see here, in this row, what happens? P is true, but Q is false. But that cannot be. It cannot be that P is true and Q is false. Therefore here we mark false for the implication. Okay? All the other columns are true.

Notice: false also implies false. Whatever your second-grade teacher may tell you — yes, false implies false; she got it from logic. Okay? And similarly, false can imply true. Meaning, if you have a false sentence, then whatever sentence you put at the end won’t change the relation of implication. It will always be true. Once the antecedent is false, it doesn’t matter what is in the consequent; the result is always true according to the material interpretation. We adopt the material interpretation of implication, which is basically this interpretation. Okay?

Okay, now of course there are many other connectives. Right? Here we presented four — five, sorry. But in principle there can be many connectives: implication, if-then, equivalence, or, and, all kinds of things. Now let’s try to think again about the question I asked earlier. Now you already have the tools to answer it. How many connectives are there altogether? 16. Right. 16.

[Speaker E] Because in each row you basically have two options, so two to the fourth is 16.

[Rabbi Michael Abraham] Exactly. Notice that each connective is ultimately just a column like this of four values, right? That’s called a connective. Give me this column — that column defines a connective. Okay? Now how many kinds of such columns can there be? We have four values; it’s a vector of length four, each component of which can take two values, true or false. So that’s basically two times two times two times two — 16. There are 16 such different columns. You can say true true true true, false false false false, false true true true, true false true true, true true false true, and so on. There are 16 such columns, no more.

Now this is a surprising result that we couldn’t have reached without formalization. We now know that no matter what words you use in language, there are no more than 16 connectives. That’s it: there are 16 connectives. Okay? Each connective is basically described by a table like the one we saw here, this table. That’s a connective. Why? Look: give me the two inputs — these are the inputs — and this is the output that comes out. Once you’ve defined all the output rows for me, you’ve defined a connective. In this case it’s the connective “or.” Okay? But you’ve defined a connective. Every connective has such a table. This table is simply a condensed presentation of five connectives, but it’s the same idea. Take these three together — that’s the connective “or.” These two together with this — that’s the connective “and,” right? These two together with this — that’s implication. And here is equivalence. Okay? This table simply gathers several tables of the kind above, that’s all.

So there are 16 such different tables, and therefore there are only 16 connectives, which is good news. Why is it good news? Because then we can hope to have a way to deal logically with propositions in language. If I want to translate every proposition in language into logical symbolism, then I want to see that I have a way to translate every proposition. And here you go: if there are only 16 possible kinds of propositions, then all I need is just 16 symbols, 16 connectives. Okay?

[Speaker F] Why is a connective defined by its truth table? Why couldn’t there be several different connectives with the same truth table?

[Rabbi Michael Abraham] Well, that’s the point — then they’re not different.

[Speaker F] Why? This says “or,” that says— maybe there could be a hundred—

[Rabbi Michael Abraham] You’re right that there can be several kinds of connectives. For example — let’s look at… wait, let’s look at… no, I don’t have a simple example here… all right, let me take another simple example. Let’s take the connective called XOR. What is XOR? XOR means exclusive-or. Yes, “either-or.” What is XOR? “Or” means it is true if at least one of its two inputs is true. Right? At least one of them is true. Only here that doesn’t hold, so only here is it false, in the fourth row. In all the other rows, at least one of the two is true. Okay? Exclusive-or says that only one of the two is true — not at least one of the two is true. It’s exclusive, meaning either this or that. All right? They can’t both be true. Either this is true or that is true. The only difference will be that here this turns into false, right? Because here both are true, and I want only one to be true. All right? So that is now the table for exclusive-or. Okay?

Do you see the connection to this table? Look, look at this. It’s simply—

[Speaker B] Negation, no? Of this.

[Rabbi Michael Abraham] Do you see the connection between these two?

[Speaker B] It’s just the opposite.

[Rabbi Michael Abraham] The opposite. So in fact, if I put a negation on this, I would get this, right? So now you see that in fact we have — but semantically, saying “it is not the case that either P or Q” is not the same as saying “P is equivalent to Q.” Those are different statements, but they have the same column. So for me, it’s the same connective. On the logical level it’s the same connective.

If someone tells me, “It is not the case that either Socrates is mortal or Yankele is mortal” — I mean with exclusive-or, yes — “it is not the case that either Yankele is mortal or Socrates is mortal,” he is basically telling me: Socrates is mortal if and only if Jacob is mortal. Notice: these are two completely different sentences from the point of view of ordinary language. But the logical translation shows us that it’s actually the same sentence in other words. Therefore I call it the same connective. Okay? Logic deals only with that. From its point of view, the connective is not tied to the semantic meaning we give those expressions, those symbols — equivalence, implication, conjunction, and so on. Rather, each of those is represented by a truth table, by a vector of four values, and that vector is the connective as far as I’m concerned. And if there is another way of saying the same thing in language, but that way also has the same vector of values, then as far as I’m concerned that is an equivalent statement — it’s the same statement. Okay?

[Speaker G] So basically the connective is indifferent to content.

[Rabbi Michael Abraham] What?

[Speaker G] Basically the connective is indifferent to content; it doesn’t care—

[Rabbi Michael Abraham] It doesn’t care what its meaning is in reality. Exactly. In the logical sense, it doesn’t care about the content. What interests it is the relation between input and output. In other words, if the same inputs give me the same output, then it’s the same connective.

[Speaker H] Even though the meaning of the connective could theoretically be very, very different.

[Rabbi Michael Abraham] Correct. That’s what I showed before, and that’s what I answered, yes, right. In Hebrew, if you say that “Socrates is mortal” is equivalent to “Yankele is mortal,” that is not the same sentence as saying “it is not the case that either Yankele or — exclusively, yes — Yankele or Socrates is mortal.” That is a different sentence semantically, in terms of the meaning of the sentences. But logically they are equivalent claims. They have the same truth table.

Now once we’ve reached this point, we understand that there are 16 connectives. That means that in principle we could now invent 16 such symbols, each symbol representing a column like this of four values, right? The symbol for equivalence represents this, the symbol for implication represents this, the symbol for “and” represents this, and so on for the 16 possible columns — we would produce 16 symbols. Once we have created 16 symbols, we can in principle formalize any sentence in language. Give me the atoms, P and Q, and I can write for you the form of the complex sentence built from those atoms. Okay?

Now of course there can also be even more complicated combinations. Say we take this—

[Speaker H] So what we’re really saying is not that there are only 16 connectives, but that we can translate all possible connectives into 16 truth tables?

[Rabbi Michael Abraham] On the logical level, that’s what is called a connective. If we look at a connective as something in language, then in language it’s true, there are more. But for me the connective is the logical issue.

Now look, for example, at this sentence. You see? This too is a complex sentence in language. All right? It’s a complex sentence in language in which I use two connectives to describe it. For example, I say: if Socrates is mortal or Yankele is mortal, then Socrates is mortal. By the way, this is of course true. This thing would be what’s called a tautology. A tautology is a column like this that is all true. In other words, no matter what happens with P and Q, the result is always true. If you build the truth table of this expression, you’ll discover that the column representing it is T-T-T-T.

All right? Look, for example, let’s take P and Q both as true, yes? Then what is P or Q? Wait, let me restore this to where it was. What is this doing here? Okay, now look. Let’s try to analyze this complex sentence. Okay? P or Q in the first row — I’m now looking at the first row. Okay? P or Q is T or T. According to the “or” table, the result is also T, right? So this is T. Now P is T, so T implies T. I go to the implication table. When the antecedent and the consequent of the implication are both T, then the implication table also gives T. Therefore the final result of all this is T. Continue checking this according to the truth tables, and you’ll discover that in all four rows you get T here. That’s what is called in logic a tautology. A tautology is a sentence that is necessarily true. It doesn’t matter whether P is true or Q is true, whether both are false — it doesn’t matter — the result of this sentence is always true.

For example, think of something everyone understands: say P or not-P. That will always be true. Agreed? It doesn’t matter what you put in place of P. Suppose P means “it is now night.” Then either it is now night or it is now day. That’s a true sentence, because it is now night. If P were “it is now day,” then I say either it is now day or it is now night. Again, it’s a true sentence. Okay? So it doesn’t matter what P is; the result is always true. That too is a tautology. All right?

Or for example, P and not-P is always false. That’s the opposite of a tautology. P and not-P is a contradiction. P and not-P is always false, whatever you put in place of P. Because it can’t be both day and not day at the same time. Or both night and not night. Right? That can’t be true, so it’s always false. And so on. I don’t want to get too deeply into—

[Speaker I] No, but if P is false and Q is true, then you’ll get true implies false, and then it’s false.

[Rabbi Michael Abraham] No — if P is true and Q false, that’s false. If P is false and Q true, that’s true.

[Speaker J] That’s true, yes, but then it implies P, which is false. Right? So—

[Rabbi Michael Abraham] That’s false, no?

[Speaker J] That can’t happen in implication. So it’s not a tautology — that’s the point.

[Rabbi Michael Abraham] P implies Q — P implies Q is false only here. Ah, you mean P implies Q here, yes? It’s also true, you’re right, so it’s not a tautology. P implies Q comes out true and P is false, so true implies false is false. I was mistaken; it’s not a tautology. You’re right. Fine. But the principle — that’s the principle, all right?

So what I want to say is that if we had 16 such symbols, each representing a column in the table, we could formalize all the sentences in language. Okay, now fortunately — at least fortunately for me — there is actually a way to represent all the connectives using a single connective. For example, let’s take the best-known connective that does this. There are a few connectives that do this, but wait — this is what’s called NAND, “not and,” okay? P and Q is “and,” and this is the symbol for negation, so it’s “not and” — NAND. Okay?

Now let’s see. P and Q is true, right? Therefore not-(P and Q) is false. Okay? P and Q is false, so its negation is true. These two are false, so their negation is true. False, its negation is true, right? This is basically the connective called NAND, not-and. This too is a connective; it’s another column that we didn’t see before.

Now it turns out — and there is a proof for this in logic — that with this connective alone, you can build all the other connectives, just with it. That is, suppose I want to build P or Q. I can take P and Q, use this connective, and then use it again together with P or together with Q, and in the end I’ll get the table I want. In other words, I can build a combination of NAND connectives that produces any connective I want. Therefore, in fact, we are in an even better situation than we thought: for the formalization of complex propositions in language, one connective is enough.

Suppose I mark this connective — I don’t know — with an asterisk, all right? Let’s say I use an asterisk. What is going on with this Hebrew? Fine. So here, for example, I mark NAND with a star, just arbitrarily. Okay? And instead of this I now write — here, I marked it like this. Okay? And now we have the NAND connective. Now if we want to build another connective, then for example we can say we have, I don’t know, this — something like this. You understand that this too is a kind of connective? Give me Q and P and it will yield some value table here.

Now it turns out that all 16 connectives I described above — all 16 of them — can be built as combinations of the star-connective in various forms of this kind. I can build all of them this way. Therefore this one connective is enough for me to use in formalizing all the sentences in language. Okay.

Now — okay, these are technical matters, not important; I’m not going to prove this here, and it doesn’t concern us right now. What I really want to say is: what do I do with these truth tables? What I want to do now is build a machine that performs this logical calculation I’ve described. For example, a machine that performs the calculation of “and,” okay? P and Q. Meaning, a machine that takes P and Q and computes P and Q for me, and gives me the result.

Notice: this is not a mathematical calculation, it’s a logical calculation. Okay? And it computes P and Q for me and gives me an output. So let me show you such a machine. Look at an electrical circuit, okay? Here we have an electrical circuit. This is a battery, say five volts for the sake of discussion, and it takes five volts to activate this bulb here. All right? This is a light bulb. Now, these are switches. There are two switches here. Okay? Now I say the following — or before I say that, what will actually happen in this circuit? If I put a five-volt battery here and a bulb here, then if the circuit is closed, the bulb will light, right? If the circuit is open, the bulb will not light. Okay?

Now here I place two switches. This is called a series connection. I place them in series, one after the other. Okay? What condition has to be met in order for the bulb to light?

[Speaker D] Both of them have to be closed.

[Rabbi Michael Abraham] Both of them have to be closed, right? And then the circuit closes. Both of them need to be closed. If one of them is open, then the bulb won’t light.

Now let’s assume that “the bulb is lit” means true, and “the bulb is not lit” means false. Okay? And P and Q will go into the switches. That is, there will be something here — P. If P is true, then this switch is closed. If Q is true, then this switch is closed. If Q is false, then it’s open. If P is false, then it’s open. Okay? Now look: if P and Q are both true, then both switches are closed and the bulb lights; the result is true, right? If P is true, then this is closed; and if Q is false, this remains open; so the bulb is not lit — that’s false. If Q is true and P is false, then this is closed and this is open; the bulb is not lit — that’s false. And if both are false, both are open; the bulb is not lit — that’s false.

So do you understand that this circuit is actually a computer? It’s a computer in every sense. It is a computer that computes the logical operation “and.” In other words, it takes P and Q as input, performs these electrical operations with them — that is, connects P and Q to these switches — and my output, my output is the bulb. If it is lit, that’s true; if it is not lit, that’s false — that’s the output. Assuming I’ve established these rules, that the input determines the state of the switches and the output is the state of the bulb, then this circuit is a computer that computes the logical operation “and.” Is that clear? It’s important to me that you be with me. Surely there are some people who are already tired of this because it’s very familiar to them, so sorry, but I want to get everyone on the same page.

[Speaker D] It represents it, it doesn’t compute it.

[Rabbi Michael Abraham] What?

[Speaker D] It represents it, it doesn’t compute it.

[Rabbi Michael Abraham] This is called computing.

[Speaker D] Okay.

[Rabbi Michael Abraham] This is an analog computer, not a digital one. In a bit I’ll talk more about the relation between them. It really is a computer. Look — just as water computes the Navier-Stokes equations, this circuit computes “and.” Okay? It simply behaves in such a way that its result is like the result of “and.” What you call “represents,” in the world of computers I call “computes.” We’ll come back to that when we look at a more realistic computer.

Now look here. This is a circuit called a parallel connection, not a series one. You see? The two switches here are in parallel with each other, not in a row one after another. What is the difference from the previous case? You can see that here it is enough for one of them to be closed in order for the bulb to light, right? Meaning, if this one is closed and this one open, then the current will flow through this lower branch, come up here, close the circuit this way, and the bulb will light. If this one is closed and this one open, then again the bulb will light, because the circuit will be completed that way. Okay? So the bulb will light. If both are closed, the bulb will also light. If both are open, that is the only case in which the bulb will not light. So what does this represent? The connective “or.” Exactly.

[Speaker B] It’s P or Q.

[Rabbi Michael Abraham] If either P is closed or Q is closed, the bulb is lit. The only case in which the bulb is not lit, giving me false, is when both are open — when both P and Q are false. Okay? Let me go back for a moment to the table, just to remind those who may have lost the thread. Look here: you see “or”? For P or Q, if P and Q are both true, that’s true. If one of them is true, that’s also true. Only when both are false is it false. That is exactly the parallel circuit we just saw. Meaning, if this switch is closed and this switch is closed, the bulb lights. If this switch is closed and this switch is open, the bulb lights. If this switch is open and this switch is closed, the bulb lights. The only case in which the bulb does not light is when both switches are open, and then the bulb doesn’t light. Therefore a parallel connection is actually an implementation — that’s the term, an implementation — of an OR gate. It is basically a logical gate that takes P and Q as input and gives out the result of P or Q as its output. And the implementation of an AND gate is the series connection. Okay?

These are basically the fundamental circuits that implement logical operations. In principle, you understand that one could build electrical circuits like these for each of the 16 connectives. Right? Give me a connective and I’ll build a circuit that ensures that if both switches are open, such-and-such happens; if one is open and one is closed, such-and-such happens; and so on. It is simply an implementation by means of an electrical circuit of logical gates. All right? That is essentially the claim.

Is this clear so far? If not, ask, because it’s important to me that you understand the principle. Right now I don’t care about the details, or about following the proofs, and there are no proofs here anyway. But I do want you to understand the idea, the mode of thinking. Okay? Is that fine? Is that clear? Okay.

So now, what I want to do next is move to a computer that performs mathematical operations, not logical ones. That is the next step toward a computer. I want a computer that won’t calculate logical operations, but will calculate for me what 16 times 84 is. Okay? Arithmetic operations, for now. How do you build such a computer? You already have all the tools. In principle you can already build such a computer — a calculator that performs computational operations of addition, subtraction, multiplication, division — let’s say addition and subtraction for the sake of discussion. In principle, you can already build such a computer.

And in order to do that, one short introduction — again, sorry to those for whom this is far too simple. In the world of computers, we usually work in what is called a binary base. That means there are only two digits: 0 and 1. In everyday life we are used to using decimal notation. There are 10 digits, including zero — 0, 1, 2, 3, 4, up to 9. There are 10 digits. But of course we can do everything in bases with two digits, three digits, four digits — it doesn’t matter. Base 2, binary, has two digits: 1 and 0. Okay?

So for example, when I write — say I write 35 — when I write the number 35 in our system, in decimal, that means I have three tens and five ones, right? Thirty-five: three tens, five ones. When I write the number 10, that means I have one ten and zero ones. Right? Now in binary, the “tens” are really twos. Two is 10, four is 100 — yes, that’s 2 squared, and so on. So for example, when I write the number 10 in binary, what I really mean is two. I have one “two” and zero ones, which is two. If I write 11, then I have one two and one one, so that’s three. 11 in binary is 3 in decimal. Okay?

Now all I really want to say — leave aside all the technical details; I think they teach this in elementary school, maybe sixth grade. If you don’t remember it, then we’re really in trouble. But never mind — the details are less important. I just want you to be convinced that it’s possible to do all of this.