The Logical Polygraph: On Liars and Human Beings (Column 200)

With God’s help

With good auspices and good fortune, we have reached column 200. The first column was written at the end of 2015, a little over three years ago, so the average pace (about sixty-five columns a year) is beyond the target of roughly one column per week that I set for myself. Well done, me.J

In column 197 we dealt with a puzzle about robbers that is related to game theory. Among other things, I discussed there the question whether game theory is supposed to describe the way human beings behave (that is, whether it is a branch of psychology) or whether it merely instructs us how it is correct to behave, given a certain utility function of the person (that is, whether it is a branch of mathematics). This reminded me of another topic in which one can also discern the tension between logical-mathematical theory and reality, from a different angle. I call it “the logical polygraph.” True, this is not a discussion of game theory but of logic, and yet there is an interesting tension here between mathematical reasoning and human behavior.

How to extract directions from a possible liar? A case of A case of doubt in which a prohibition has been established. (an established doubt)[1]

Once upon a time, when I was teaching science-oriented teenagers at the university, we dealt with logic and its meaning. I presented the youngsters with a short article by Yael Cohen (of the Hebrew University), called “How to Get the Truth out of Any Liar”.[2] The author proposes there a logical algorithm that makes it possible to extract the truth from any person, regardless of whether he is a liar or a truth-teller.

She begins the discussion with the well-known puzzle[3] about a man walking on the road to heaven.[4] He comes to a fork in the road, one path leading to heaven and the other to hell. At the entrance to each path stands a guard, one of whom is a liar and the other a truth-teller. The traveler, of course, does not know which is the liar and which is the truth-teller, and he may ask one of them one yes/no question in order to find out which way he should turn. What should he ask?

The answer is well known. He should really ask one of them:

A. What would the other one answer if I asked him whether the road on the right leads to heaven?

Once he receives an answer, he should of course do the opposite. Check and you will see that regardless of the nature of the respondent (whether he is a liar or a truth-teller), the traveler who acts this way will always arrive in heaven.

How to extract the truth from a possible liar? A case of A case of doubt in which no prohibition has been established. (an unestablished doubt)

Another version (this was probably Goodman’s original version) speaks of a tribe composed of truth-tellers and liars. One of them stands guard at the fork, and the traveler does not know whether he belongs to the liars or the truth-tellers. He must ask him one yes/no question and discover the correct road to heaven. What should he ask?

Here too the answer is quite similar:

B. What would you answer me if I asked you whether the road on the right leads to heaven?

If you check, you will see that here he must do exactly what he is told (not the opposite, as in the previous example), and regardless of who the respondent is (whether he is a liar or a truth-teller), if he acts accordingly he will reach heaven.

How to extract the truth from a possible liar without counterfactual statements

These questions, however, use counterfactual statements (counterfactual: what would happen if…), and analytic philosophers think such statements are problematic. Goodman therefore asks whether it is possible to extract the truth from the guard at the fork by means of a single question that does not use such problematic statements. He is looking for a question based only on a compound proposition and logical connectives. It turns out that here too the answer is yes. One may ask the guard the following question:

C. Are you a liar if and only if the road on the right leads to heaven?

Here the traveler must do the opposite of what he is told, in the following sense: if the respondent answers “yes” he should turn left, and if he answers “no” he should turn right.

The analysis goes as follows: this proposition (= sentence) is composed of two atomic propositions (= clauses)[5]: (A) “You are a liar.” (B) “The road on the right leads to heaven.” The connective that combines them is equivalence (if and only if, denoted by =).[6] Therefore the formalization of the sentence is: A=B. The definition of the equivalence connective in logic is that the compound proposition is true if and only if the two clauses have the same truth value (either both false or both true).[7]

I ask the guard whether A=B. The answer to that question is “yes” if the truth value of the compound proposition is true, and “no” if its truth value is false. But of course, if our guard is a liar, he will answer the opposite of what he ought to answer (if the correct answer is “yes,” a lying guard will answer “no,” and vice versa).

To see why such a question extracts the truth from any person, we must divide the discussion according to whether the respondent is a liar or a truth-teller:

• The guard is a liar. In that case proposition A is true. Therefore, if the road on the right leads to heaven, then proposition B is also true, and therefore the truth value of the compound proposition is true (because the truth values of the two clauses are identical). But since he is a liar, he will of course answer “no.” If the road on the right does not lead to heaven, then proposition B is false, and therefore the whole compound proposition is false. But as a liar he will of course answer “yes.” Therefore, if he is a liar, then when the answer is “no” the traveler should turn right, and if the answer is “yes” he should turn left.
• The guard is a truth-teller. In that case proposition A is false. If the road on the right leads to heaven, then proposition B is true, and therefore the whole compound proposition is false (because the truth values of the two clauses are not identical). The truth-teller will of course answer “no.” And if the road on the right does not lead to heaven, then proposition B is false, and therefore the truth value of the compound proposition is true, and the respondent, being a truth-teller, will answer “yes.” Therefore in this case too the traveler should turn left if the answer is “yes,” and right if the answer is “no.”

In both cases the traveler must do the same thing. Therefore, regardless of the nature of the guard who was asked, if the answer is “yes” the traveler should turn left, and if it is “no” he should turn right. So it does not matter what kind of guard stands before him: our bewildered traveler has a clear way to extract the truth from him, and this time even without using counterfactual statements.

But perhaps the guard does not know logic?

And what if the guard does not know logic and does not understand the meaning of the equivalence connective? An ordinary person would not even think to answer such a question at all. He simply would not understand it, and would probably content himself with delivering a punch aimed directly at the dropped jaw of our innocent traveler, who is bothering him with meaningless questions.

Of course, by the same token we could object that perhaps the guard does not know Hebrew, or that he does not know what the word ‘heaven’ means. These are not troubling questions, because this is only a technical problem. At most we will teach him Hebrew and the terms being used, and the logical problem is well defined. If so, the same applies to the problem of knowing logic. In such a case, our traveler should teach the guard a lesson in logic and clarify for him the meaning of this connective, and then return and ask him. For purposes of the logical discussion, the assumption is that our guard understands what is being said to him, that is, he knows the language. Just as we assume that he speaks Hebrew, there is no impediment to assuming that he also understands the meaning of the equivalence connective. Later we will see the importance of this question for our purposes.

And what if the guard is inconsistent?

So far we have dealt with consistent fellows, who either always lie or always tell the truth. With that crowd it is easy enough to manage. It turns out that an inconsistent fellow is a much harder nut to crack (see column 9). Let us now assume that the son of that traveler arrives at the same fork, and there stands one of the descendants of the same tribe his father encountered. Now we have a guard at the fork, and the assumption is that, because of the wonders of genetics, he may be a more complex character, that is, a person without a moral backbone. The fellow may very well be neither a liar nor a truth-teller, but someone who decides anew each time whether to tell the truth or to lie. Can the correct instruction be extracted even from such a dubious character with one yes/no question?

At first glance one might use the last question (C), since as we have seen, whether he lies in relation to it or tells the truth, his answer should be the same. But that is not precise. Think about the truth value of proposition A in our case. The fellow is neither a liar nor a truth-teller. The problem is that here there are three possibilities, not only two as before (either he is a liar, or he is a truth-teller, or he is inconsistent).

What can be done is a slight variation on question C, namely to ask him:

D. Will you lie in answering this question if and only if the road on the right leads to heaven?

Here we did not ask whether he is a liar but whether he is lying now. In that situation there are only two possibilities (and not three): either he is lying now or he is not. It does not matter to me whether over the course of his life he is consistent or not, because I am not asking him that.[8]

The problem of the logical polygraph

Cohen raises the question there: why not use such a formulation to extract the truth from any liar? For example, a woman who suspects that her husband is being unfaithful to her can ask him:

E. Will you lie in answering this question if and only if you were unfaithful to me?

The police, too, who investigate criminals, could use the logical polygraph presented here. Suppose a person is being investigated on suspicion of theft. All that one needs to ask him is:

F. Will you lie in answering this question if and only if you stole this object from so-and-so?

If he answers “yes,” it is clear that he did not steal it, and vice versa.

So why do we go to such lengths building polygraphs (which today are not recognized as legal evidence), conducting complicated interrogations, and searching for evidence? We need only require the accused to answer one question in the interrogation room, and that is all. No torture is needed for this either. All that is required is to deny him the right to remain silent, that is, to establish by law that if he does not answer, his silence will be used against him and he will be convicted.

A first note from Jewish law[9]

So that, God forbid, we not sin through neglect of Torah study, let me make a note from Jewish law (Turn it over and over, for everything is in it. — turn it over and over, for everything is in it). We should discuss whether such an investigative method runs afoul of the rule A person does not render himself wicked. (“a person does not render himself wicked”), since we are convicting him on the basis of his own testimony. First, even if that is correct, then at least in civil law (monetary law) this could be used, for there A litigant’s own admission is equivalent to a hundred witnesses. (a litigant’s admission is like a hundred witnesses).

But it seems to me that even in criminal law there is no problem in a case like ours. In this case, the suspect’s answer to the question reveals the truth to us as indirect evidence, and not because of testimony issuing from his mouth that incriminates himself. After all, he can even lie, and we would have no problem (even if he lies, we would discover the truth this way). There is no prohibition in Jewish law against using evidence that comes from a person’s mouth, only against testimony in which he incriminates himself. That is not the case here. Incidentally, it follows from this that even in civil law such an admission would not have the status of a litigant’s admission that is like a hundred witnesses (at most it would be evidence like a presumption), for this is not an admission but evidence extracted from mere speech.

Generalization

In fact, we have here a means of extracting the truth from any person. If you want to know whether X or not, you must ask the person who has that information:

G. Will you lie in answering this question if and only if X?

Regardless of whether he decides to lie to me or to tell the truth, if he answers “no” I know that X is true, and if he answers “yes” I know that X is not true. Of course, as we saw above, we must make sure that the respondent knows the language of logic, in particular the meaning of the equivalence connective. If he does not know it, we will teach him a short course in logic. That is still cheaper and more efficient than maintaining complicated, expensive, and not necessarily reliable investigative apparatuses. Success is guaranteed.

The interesting question is how it could be that one can extract any information whatsoever from any person by purely logical means. I assume you will agree that this is utterly implausible.

Self-reference

But before we get to that question, note that there is nonetheless a problem here. In fact, this is a claim that refers to itself, and therefore the analysis of it is more complex than the one we conducted above. To see this, we have to translate the answer “yes” into a declarative proposition. A person who answers question G with “yes” is really asserting:

H. I am lying when I assert H if and only if X.

This is already very much like the liar paradox, for he is claiming that he is lying in this very claim. We have here a self-referential claim (see columns 195–196).

Similarly, when a person answers “no,” he is really asserting:

I. I am lying when I assert I if and only if not X.[10]

Here too we have self-reference of the liar-paradox type. The skilled reader who made it through columns 195–196 can surely already construct the parallel anti-paradoxical claims.

Analysis

So how does one analyze this complicated claim logically? What does one do with the self-reference? Cohen proposes an analysis by means of a truth table. To do so we must define the following proposition,[11] which we shall call J:

J. I am lying when I assert H.

When one examines proposition J, one immediately sees that there is a connection between the truth value of J and the truth value of H: if H is true, J is false, and vice versa.

Now let us construct a table representing the truth values of the relevant clauses and propositions (False – false, True – true):

H: (X)=(J)[12]	X	J	H
F	T	F	T
T	F
T	T	T	F
F	F
(4)	(3)	(2)	(1)

How was the table constructed? The first two columns (1-2) survey all the possibilities that exist here. H can be true or false. But given the truth value of H, the truth value of J is determined, since as we saw they must be opposites. In each of these two possibilities we can assume the possibility that X is true or false, and that is column (3). The truth value of the last column, (4), is determined by the equivalence between X and J, that is, between columns 2 and 3: if their truth values are identical, there is a T there, and if not, there is an F there.

But now we must remember that column (4) is none other than H, and therefore its truth value must be identical to column (1). The meaning of this is that in column (4), out of the two possibilities in each row, only the bottom two are consistent; in the other two cases we arrive at a contradiction with column (1). Therefore it is clear that in both cases the truth value of X is F.

This is one way to analyze the sentence despite the self-reference it contains. Unlike the liar paradox, here one can assign a truth value to this proposition (despite the fact that it contains an element of self-reference). Thus we have proved here that if the respondent asserts claim H, whether he is lying or telling the truth, the truth value of X is F, that is, X is false. We extracted the truth from him regardless of his moral character. The same analysis shows that when the respondent asserts I, X must be true, again regardless of his moral character.

So why indeed not use this logical polygraph, whose reliability is absolute, instead of building investigative mechanisms and risking failure, cost, and various other human complications?

Can there be a logical polygraph?

The proof that if the respondent answers “yes” then X is necessarily false relied on the fact that if X were true, the respondent would become entangled in a paradox, since in that state the very same proposition receives different truth values, in columns (1) and (4) of the table. In the background it is presumably “obvious” to us that the respondent very much does not want to become entangled in a paradox. But why, really, not? What prevents him from answering “yes” even if X is true? The fact that this creates a paradox is a logical fact, but there is nothing that would prevent him from moving his lips in such a way as to say the word “yes.” Will his lips fail to move? When we hear such an answer we will go into a loop, but that need not really concern our suspect. He does not necessarily sympathize with his investigators, and he does not necessarily want to save them (or himself) from becoming entangled in paradoxes. If that is what will save him from prison, then of course that is what he will do.

Above I asked what one does if our suspect does not know logic. I answered that this is only a technical problem, since he can be taught the relevant language. But now I am speaking about a suspect who knows logic and understands the meaning of the equivalence connective, and still gives an answer that entangles us and himself in a paradox merely in order not to be caught for the offense. This is already a substantive problem and not merely a technical one, for that is precisely our concern — that the offender will say something in order not to be caught red-handed. That is what we wanted to prevent, not to give him the opportunity to say something that conceals the truth from us. In practice, of course, we did not succeed in that. A logical polygraph cannot do the job, because logic determines truth values and relations between claims, but not necessarily what flesh-and-blood human beings will do or say. Even if they know the logical theory and are supremely rational, they can still say what is not expected of them by the logical analysis. On the contrary, precisely if they are rational, they will say exactly that in a case where they are guilty or want to hide the information.

And one more note from Jewish law

One may ask whether, when the respondent committed the offense and nevertheless answers “yes” (in order to cause us to think that he did not commit the offense), he has violated Keep far from a false matter. (“keep far from a false matter”). There is room for the claim that he has not. He did not answer the question whether he committed an offense. He answered a different question, and only by doing so caused us to think that he had not committed the offense. Therefore he did not really lie in his testimony.

To be sure, one might argue that with respect to the question itself he did lie, for if he committed the offense, then the answer to question H should be “no,” whereas he answered “yes.” But that too is not precise, for that is not the answer to question H itself, but rather the answer to H that does not entangle him and us in a paradox. As we saw in connection with the claim of the liar paradox, it is doubtful whether the question itself has a clear meaning (see columns 195–196). Beyond that, it is clear that there is no lie in testimony here, but at most an ordinary lie. On the face of it, there is no prohibition in Jewish law on an ordinary lie, only in a religious court.[13]

Incidentally, this note from Jewish law, like the previous one above, is closely bound up with the distinction I made in the previous section between logic and a person’s actual behavior. Self-incrimination and lying are concepts that deal with what a person does and with the meaning of his claims, whereas here we are dealing only with the logical analysis of his claims and with what follows from them. As we saw, it is important to distinguish between these two planes.[14]

Summary: back to Buridan’s ass and the question of rationality

We have seen here another example of the fact that logical analysis does not necessarily help us understand human beings and predict their behavior or draw conclusions from it. The reason is that a person is not necessarily rational and logical, and therefore analysis of his claims does not necessarily teach us about his position and what he thinks. Incidentally, as I already noted, in fact here he may not be logical, but he certainly is rational. A rational suspect should say what will save him from conviction, and that is exactly what he does.

This, of course, brings us back to column 196, to the discussion of Buridan’s ass and the question of rationality. There I distinguished between two kinds of rationality: causal rationality (a person who performs only actions for which there is a reason to perform specifically them and no other alternative) and Buridan rationality (performing actions that bring maximal benefit). Similarly here too I distinguish between logical rationality (asserting only what is consistent and non-contradictory on the logical plane) and utilitarian rationality (asserting what brings me maximum benefit).

[1] Jewish law distinguishes between two kinds of doubt: when there is certainly one forbidden piece among two pieces and we do not know which one is forbidden — this is A case of doubt in which a prohibition has been established.. When there is one piece before us and we do not know whether it is permitted or forbidden — this is A case of doubt in which no prohibition has been established.. The first doubt is considered more severe (for example, in Jewish law only for it does one bring a suspended guilt-offering). In our case too, the doubt regarding the guard can be of both kinds: if there is certainly one guard who tells the truth and one liar, and we do not know which is which — this is A case of doubt in which a prohibition has been established.. One guard whose status we do not know, whether he is a liar or a truth-teller — this is A case of doubt in which no prohibition has been established.. Incidentally, from a logical point of view, specifically the second doubt is harder to handle; see below.

[2] Iyyun 30, issue 1 (Tevet 5741), pp. 41-47.

[3] The author is the philosopher and logician Nelson Goodman. An early version of it appeared in the Boston Post in 1931, and after several decades in his book Problems and Projects.

[4] In her version, unbeliever that she is, he went to the capital city and not to heaven. But the familiar version deals with heaven, and I hold fast to the tradition of my ancestors.

[5] In the next column I will explain more fully the difference between a proposition and a clause, and its implications for Jewish law and otherwise.

[6] It is sometimes called the identity connective. Equivalence and identity are the same thing on the logical plane (see the next note).

[7] This is of course not the usual definition of identity between two propositions, since identity between two propositions is connected to their contents. Even if it is daytime outside right now, the sentence “It is daytime outside right now” (which is true) is not identical to the sentence “Netanyahu is the prime minister of Israel,” even though both are true. But in logic one uses definitions that depend only on the truth values of the propositions and not on their contents. So too with implication (which is usually defined materially: A->B means that it cannot be the case that A is true and B is false), and so too with equivalence, for A=B is nothing other than double implication: A->B and B->A.

[8] Incidentally, it is interesting to think about the version that includes counterfactual statements. It is very easy to see the problem in answering it in this case (the respondent will not be able to answer what he would do if he were asked a different question, since he makes decisions locally, each time differently).

[9] I raise here an interesting point with quite a few implications, and I intend to expand upon it in one of the coming columns.

[10] This is the negation of claim H, if one takes into account the meaning of the equivalence connective.

[11] In fact this is not really a clause but a (compound) proposition. In the next column I will address this distinction and its implications for Jewish law and otherwise.

[12] When you read claim H you will see that it is nothing other than an equivalence between X and J.

[13] See, however, the words of Rabbeinu Yonah in Sha’arei Teshuvah where he elaborates and details the prohibition on lying, and also the article by Rabbi Yuval Cherlow, “Keep far from a false matter.: A Brief Digest of the Laws of Lying,” in Tzohar and also online.

[14] As stated, I will devote a column to this later on.