A Logical Look at the “Trei u-Trei” Sugya (Column 319)

With God’s help

Disclaimer: This post was translated from Hebrew using AI (ChatGPT 5 Thinking), so there may be inaccuracies or nuances lost. If something seems unclear, please refer to the Hebrew original or contact us for clarification.

I’m on a roll from the previous column, so I’ll now take up another sugya and look at it through a logical lens. By way of preface: in column 195 I defined logical structures of paradox and anti-paradox, and in column 196 I brought halakhic examples for these two structures. Here I’ll briefly reprise those definitions and present them somewhat differently, in order to apply them to the “Trei u-Trei” sugya.

What is a proposition?

Aristotle already taught us that not every sentence is a proposition. When I say that it is daytime outside, that is a proposition. It can be true or false, and we examine this by comparing the content of the proposition with the state of affairs in the world that it describes. A proposition, according to Aristotle, is a sentence that describes some state of affairs, that is, it asserts something about the world. Therefore it is judged in terms of truth or falsity. The proposition that I am now inside a cloud is certainly a proposition, since it asserts something about the world. Admittedly it is a false proposition (you can see this if you compare it with the state of affairs it describes), but it is still a proposition.

By contrast, sentences like “What time is it?” (a question) or “Do X!” (a command) are not propositions. They do not assert anything about the world but perform a different linguistic act. One cannot say of such sentences that they are true or false, and therefore they are not propositions. In fact, it’s more accurate to say that because they are not propositions, one cannot say of them that they are true or false.[1] For our purposes, a proposition is a sentence to which one can assign a single truth value from two possibilities: true or false.

Paradox

There are sentences that might seem to assert something, and yet it is impossible to assign them a truth value of true or false. One of the best known is the Liar Paradox,[2] whose simplest formulation is the following:

(א): Sentence (א) is false.

If we assume this sentence is true, then it itself turns out to be false; therefore its claim that it is false is true, and therefore it is true, and so on ad infinitum. One cannot assign the value ‘true’ to such a sentence, nor the value ‘false’.

There is a more complex, but equivalent, formulation which, for what follows, we’ll denote as structure A:

(א): Sentence (ב) is true.

(ב): Sentence (א) is false.

Here, too, the calculation is similar: if we assume sentence (א) is true, then (ב) is also true. But (ב) says that (א) is false, and so on ad infinitum. Clearly, if we swap the names of the two sentences we get exactly the same structure (which we’ll denote structure B); that is, we have two dual structures that represent a paradox.

Thus, a paradox is a sentence (or a two-sentence structure) to which no truth value (true or false) can be assigned.

Anti-paradox

There is another kind of sentence that on its face looks like a proposition, but on closer inspection is not exactly such. Consider the sentence:

(א): Sentence (א) is true.

If we assume it is true, then it really is true and all is well. So why isn’t it a proposition? Because even if we assume it is false, it turns out to be really false, and again all is well. Thus, this sentence too cannot be assigned a unique truth value; but it differs from the paradoxical structure in that there it was impossible to assign any truth value, whereas here one can assign to this structure either of the two truth values: it can be true or it can be false. We cannot determine which of the two obtains (not because of our inability, but because the sentence has no unique truth value). I called this an anti-paradox.

Does the anti-paradox also have an equivalent formulation as a two-sentence structure? It seems so. Consider the following formulation, which will be denoted structure C:

(א): Sentence (ב) is true.

(ב): Sentence (א) is true.

In this structure one can assign the value ‘true’ to both sentences, and also the value ‘false’ to both. Both assignments are consistent, and therefore one cannot decide between them. The structure is true or false at our discretion. In this it differs, as noted, from the paradox.

Now consider the following structure (denoted D):

(א): Sentence (ב) is false.

(ב): Sentence (א) is false.

Here the situation is slightly different but in a certain sense still similar. One can say that sentence (א) is true and (ב) is false, but also that sentence (א) is false and (ב) is true. Both possibilities exist and are consistent, and therefore here too the structure admits two opposite truth-value assignments (that is, a vector of truth values for the two sentences). Hence, such a structure can also be treated as a kind of anti-paradox. We have two dual structures that represent an anti-paradox.

Graphic representation

After translating the paradox and the anti-paradox into dual structures, it becomes easier to represent each such structure graphically (see column 195). There are essentially four possible structures: two paradox structures (A and B, which are entirely equivalent up to swapping the labels of the sentences) and two anti-paradox structures (C and D, which, as we saw above, differ from one another). We will represent them via M. C. Escher’s famous drawing.

His original drawing is called “Drawing Hands” (1948):

We see two hands drawing one another. We will treat the hand that draws its counterpart as expressing a sentence that affirms its fellow, such as: (א) Sentence (ב) is true. According to this reading, Escher’s drawing represents structure C, i.e., the first type of anti-paradox.

By the same token, one can consider a sentence that denies its fellow, such as: (ב) Sentence (א) is false, and represent it by a hand erasing its counterpart. Consider now the following drawing (a paraphrase of Escher’s drawing that I found in an online T-shirt ad):

According to the reading I suggested here, this drawing represents the second type of anti-paradox (structure D).

It is now also clear how to represent structures A or B graphically, i.e., the paradox. The drawing should be a hand erasing its counterpart that is drawing it (one hand with an eraser and the other with a pencil). Unfortunately, I did not find such a drawing online, and I’ll leave you to imagine it yourselves.

We can now move to the sugya of Trei u-Trei.

The sugya of Trei u-Trei[3]

As is known, two witnesses constitute the ultimate proof in halakha. The strength of such proof is considered absolute, since halakha rules that “two are like a hundred” (i.e., two witnesses equal a hundred). Admittedly, there are proofs that nevertheless override witnesses, such as “the murdered man comes on his own feet” (which is really part of the court’s own direct sight, and this is the Talmudic rule “hearing is not greater than seeing”), admission by a litigant (hoda’at ba’al din), be-yado (“it is in his power”), shavyei anafshei ḥatikhah de-issura, or “the mouth that forbade [is the mouth that permits]” (at least according to some Rishonim),[4] but for our purposes what matters is the absoluteness of two witnesses as proof.

It is no wonder that when testimonies of two contradicting sets of witnesses collide—one says Reuven murdered and the other says he did not—the halakha sees this as a pathological state that is almost impossible to resolve. Consider what could possibly decide such a clash: even if another hundred witnesses came for one side it would not help, for then we would have 102 against 2, but “two are like a hundred.” If so, certainly no other kind of proof will help, for the same reason.[5]

Discussion of Trei u-Trei appears in several sugyot throughout the Talmud and is divided into two main topics: the status of the subject of the testimony and the status of the sets of witnesses. Suppose set A testifies that Reuven murdered Shimon, and another set B testifies that he did not. The first discussion concerns Reuven’s status—do we consider him a murderer or not (do we maintain him in his chazakah or not). This is addressed mainly in Yevamot 30b–31a, where there is debate whether Trei u-Trei is a biblical doubt (sefeika de-oraita, i.e., we do not maintain his chazakah) or a rabbinic doubt (sefeika de-rabbanan, i.e., biblically we do maintain his chazakah; the Rishonim dispute whether rabbinically as well). But in most sugyot the Trei u-Trei discussion addresses the status of the sets of witnesses themselves. There too the question is whether we maintain them in their presumption or not. As we shall immediately see, some Rishonim link these two discussions.

The Amoraic dispute within the second discussion

In Shevuot 47b the Amoraim dispute the law of the two sets of witnesses:

It was stated: Two sets of witnesses that contradict one another—Rav Huna said: this one comes on its own and testifies, and that one comes on its own and testifies; Rav Ḥisda said: “Why do I need lying witnesses?”

Since there is contradiction between the two sets, it is clear that one of them is lying. Yet each set has a presumption of validity, and one of them is apparently telling the truth. The question is what to do in such a case. According to Rav Huna, each set retains its presumption of validity and therefore may testify in future cases. Rav Ḥisda disagrees and holds that both sets have lost their presumption of validity and neither may testify in court in the future.

One could have said that for each set there is a Trei u-Trei collision, and the dispute between Rav Huna and Rav Ḥisda is whether in such a case we maintain its chazakah or not. If so, this discussion is precisely the discussion in Yevamot cited above: whether in Trei u-Trei we maintain the matter in its presumption or not.

Note that de facto virtually all decisors rule like Rav Huna, that we maintain both sets in their presumption.

The view that identifies the two discussions

In Ketubot 21b–22a the Gemara discusses a case of Trei u-Trei concerning the fitness of a judge (two say he is fit and two say he is a thief, hence unfit to judge). The Rishonim dispute the meaning: according to Rashi, in such a case we do not maintain the judge in his presumption of fitness (as per the view that Trei u-Trei is a biblical doubt), but according to Ba’al ha-Maor and R. Ḥananel he remains in his presumption of fitness (as per the view that Trei u-Trei is rabbinic). Apparently this is a discussion of the first type (treated in Yevamot: is Trei u-Trei a biblical or rabbinic doubt), since the question concerns maintaining the subject of the testimony in his presumption, not the status of the witnesses.

Yet Tosafot there (s.v. “Trei u-Trei,” 22a) cite Rashi’s view and challenge it:

“It is two against two”—Rashi explained that when they say ‘he is not a thief,’ it is two against two, and he is not thereby certified. This is difficult, for here we proceed according to Rav Huna, who holds (Shevuot 47b) that ‘this one comes on its own and testifies,’ so we should say: maintain the man in his presumption.”

Tosafot ask why we don’t maintain the judge in his presumption of fitness, since de facto we rule like Rav Huna, who maintains the sets of witnesses in their presumption.

At first glance this is puzzling, for the question here concerns the subject of the testimony (the judge’s fitness), whereas Rav Huna vs. Rav Ḥisda concerns the fitness of the witness sets. On the face of it the sugyot treat two different questions. Tosafot assume that the dispute about the witness sets also applies to the subject of the testimony: whoever maintains the witnesses in their presumption (Rav Huna) will also maintain the subject in his presumption, and vice versa.

Tosafot then continue and argue that in our case even Rav Ḥisda, who does not maintain the witnesses in their presumption, must maintain the judge in his presumption:

“Moreover, according to Rav Ḥisda who said (ibid.) ‘why do I need lying witnesses,’ he would concede here that this man is automatically certified; for if one of these sets came to testify about some man that he is disqualified, then according to Rav Ḥisda, who considers them lying witnesses, she would not be believed—even though no one contradicts her—and all the more so where there are witnesses who contradict, that they are not believed.”

They argue that if one set came a year later and testified alone that a judge is disqualified, then according to Rav Ḥisda we would not accept its testimony. If so, here too—where there is another set that testifies that the judge is fit—we certainly would not accept its testimony.

This seems quite reasonable, but Tosafot ignore the obvious distinction between that later testimony and the original, colliding testimonies. The invalidation of the witness sets, according to Rav Ḥisda, concerns future testimonies, after it has become clear that they lied. Regarding the testimony in which they clash with one another, in that very testimony they are not yet invalid. It seems Tosafot assume that if one set is lying, it thereby immediately becomes invalid. In other words: they are unwilling to distinguish between falsity in testimony and disqualification of witnesses. In any case, for Tosafot the discussion about the status of the witnesses is the same as the discussion about the subject of the testimony.

The difficulty with Tosafot’s approach

As noted, this linkage is very puzzling. First, in the Gemara itself these are presented as two entirely separate discussions. None of the sugyot links them. From another angle, according to Tosafot there could be no view that Trei u-Trei is a biblical doubt (i.e., that in Trei u-Trei we do not maintain the subject in his presumption), since that would not fit either Rav Huna or Rav Ḥisda—given that Tosafot argued both would agree to maintain the judge in his presumption.

Beyond this primarily textual problem, even conceptually the connection between the discussions is unclear: in the discussion about the subject of the testimony, there is a collision between two sets, which is the ordinary case of two versus two. We are in doubt whether Reuven murdered or not, or whether the judge is a thief or not. But when discussing the status of the sets themselves, with respect to each set there is no Trei u-Trei, since witnesses cannot testify about themselves. For example, if two witnesses come and testify that Reuven murdered, and two other witnesses come and testify that the first set is invalid as witnesses, the first set is disqualified. There are two witnesses disqualifying them, and their own testimony about themselves is of course inadmissible. By that logic, so too in the discussion of the witnesses’ fitness in the case of Trei u-Trei: for each set there are only two witnesses disqualifying it, and there are no two who certify it (for witnesses are not believed about themselves). It would seem that each set should be certainly invalid (not merely out of doubt), similar to Rav Ḥisda—though according to my analysis this would yield certain disqualification, whereas his formulation sounds like disqualification out of doubt.

Explaining Rashi’s view

Rashi, as noted, holds that even if we rule like Rav Huna that we maintain the sets in their presumption, still with regard to the subject of the testimony it may be that we do not maintain him in his presumption. It is evident that for him these are two independent disputes. The likely explanation, on his view, is as follows: in Trei u-Trei we do not have a set of witnesses disqualifying any set, because the disqualifying set itself is disqualified by the other set. If we disqualify both sets, we are doing something self-defeating: on the one hand we say set A is invalid, and on the other we use that very set to invalidate set B, and vice versa. Therefore Rav Huna prefers to leave both in their presumption (for to remove something from its presumption we need testimony, and here we have none). Still, with regard to the subject of the testimony, even Rav Huna might agree that we do not maintain him in his presumption, because here we have two testimonies by two fit sets (at that time they have not yet been disqualified). This is a collision of two absolute proofs, neither of which has been invalidated, and therefore the situation is not juridically a “doubt” but rather two colliding certainties. Thus, contrary to Tosafot’s assumption, there is no impediment—according to Rav Huna—to distinguishing between the subject of the testimony and the status of the witnesses.

By contrast, for Rav Ḥisda we do not certify either set. It seems that for him, a set of witnesses about whom there is doubt whether they are telling the truth is not a set of witnesses. The absoluteness of the two-witness proof demands that we have certainty about that testimony. Testimony shadowed by doubt—even if not proven false—still lacks certainty. Such testimony cannot be deemed testimony (this is the language of Rav Ḥisda: “Why do I need lying witnesses?”—to remain in doubt I don’t need witnesses. Witnesses are meant to eliminate doubt).

An explanation via anti-paradox

We can understand this Amoraic dispute if we view the situation as a whole, rather than discussing each set separately. Set A is disqualified by set B, and there is no set certifying it (since they are not believed about themselves). The same for set B, which is disqualified by set A. But now note that if set A is disqualified, it cannot disqualify set B, and vice versa. That is, the two claims (“set A is invalid” and “set B is invalid”) cannot both be admissible together. Of course, it is equally impossible that both are fit. What we have is one of two possibilities: either set A is fit and the other is invalid, or set B is fit and the other is invalid. This is precisely the situation of the anti-paradox of type D described above.

Note that my claim here does not concern the content of the testimony. Clearly we cannot accept both testimonies together, since they contradict. Regarding the content, only one set is telling the truth and the other is lying. But here our concern is the fitness of the sets for future testimony (not this testimony). We saw above that Tosafot indeed identify the two questions, but on their face they are different. In such a case it is not enough to decide that one of them lied, for even if there is doubt regarding each, each still has a presumption of fitness. Nevertheless, because the disqualification of set A comes from the mouth of set B, which itself was disqualified by set A—and likewise for set B—a situation is created of the anti-paradox of type D (each of the two sentences asserts that the other is false), represented by the T-shirt ad drawing (each hand erasing the other).

We have seen that in the discussion about the sets of witnesses we have a doubt structured differently from the doubt regarding the murderer/judge himself (the subject of the testimony). Regarding the subject, we are in doubt which of the two sets is telling the truth and which is lying (a doubt regarding the content of their statements). On the face of it this is a paradoxical situation, since we cannot say Reuven murdered and also not say that he did not murder (remember: two witnesses constitute an absolute proof that cannot be neutralized). But regarding the status of the sets, we saw (unlike Tosafot) that there is no Trei u-Trei collision. Here the structure is different: we have an anti-paradox, in which there are two possibilities: either set A is fit and the other is invalid, or set B is fit and the other is invalid.

In light of the above, we can understand the Amoraic dispute differently:

• Rav Ḥisda cannot use chazakah regarding both sets of witnesses, because the chazakah leads to a state that is not one of the two possibilities between which we are in doubt. Recall that in an anti-paradox there is the possibility that A is fit and B is invalid, or that B is fit and A is invalid. The possibility that both sets are fit is not one of the two options between which we are uncertain, and therefore we cannot rely on chazakah and rule so. What remains is that both are of doubtful status (they have no status), but the rule is that if there are witnesses about whom there is doubt, their testimony is not accepted (they are not invalid per se, for invalidating both A and B is also not one of the two possible options). According to Rav Ḥisda, we do not decide regarding the witnesses’ fitness. By contrast, in the discussion about the subject of the testimony, we are in doubt whether the judge is fit or unfit, and the chazakah says he is fit. Here there is no impediment to maintaining his presumption, because, as we saw, it yields a result that matches one of the two possible options.
• Rav Huna maintains by chazakah even in a state of anti-paradox, because in his view a halakhic ruling based on chazakah need not conform to one of the very options between which we are uncertain (this is the discussion of chazakah in terei de-satrei, e.g., the case of the “two paths”—see Ketubot 27a and parallels). In a state of paradox (i.e., concerning the subject of the testimony), it may be that he also maintains by chazakah (as Tosafot assume). But it may also be that he does not, since this is not a doubt between two states; rather, neither state is possible (this is a paradox), and perhaps in such a state we do not apply chazakah.

Tosafot, who link the two disputes, are unwilling to distinguish between the truth of the witnesses’ words and their fitness or invalidity. For them what matters is only the truth value of their claims. Therefore they identify the discussions about the subject of the testimony and the fitness of the witnesses. In their view, the question is whether they are telling the truth, not whether they are fit. Therefore they do not see Trei u-Trei as an anti-paradox. No set of witnesses addresses the other (i.e., no one claims the other is invalid). Both sets address the subject of the testimony, not the fitness of the opposing set, and the question of truth and falsehood in their statements is a by-product that arises for the judges. Hence, according to Tosafot there is no anti-paradoxical structure here, and they identify the two discussions.

Additional cases

We have seen that Trei u-Trei is a structure of the second type of anti-paradox, D. The first type of anti-paradox (structure C, where each set says the other is telling the truth) is not relevant in the legal context. In such a case we would have a set of four witnesses about the event, and that is that (given the presumption of fitness, we do not fear the logical possibility that they are all lying). Even if the two sets each testified about the other that it is telling the truth (and not about the subject of the testimony), this has no legal-halakhic significance, since in halakha there is no need for testimony about the fitness of witnesses. They enjoy a presumption of fitness unless proven otherwise.

A legal-halakhic structure parallel to the Liar Paradox (schemes A or B) could arise if one set says the other is lying and the other says the first is telling the truth, or vice versa (a third version of a hand erasing a hand drawing it). In such a case it is likely that the legal result differs from the logical result. We would accept the testimony of the set about whom it is testified that it is telling the truth, for everyone agrees it is telling the truth and it has a presumption of fitness without testimony against it. And the other set would be disqualified because there is testimony that it is lying. But such a case cannot arise through the subject of the testimony. If set A testifies that Reuven murdered, what would set B have to testify in order to generate the anti-paradox C? This could occur only if the two sets address one another and not some third subject of testimony.

This leads to another interesting implication: when two sets of witnesses testify head-on about one another that the other is lying (i.e., is invalid). If we maintain one by its presumption, then the content of its testimony is that the other lied, and conversely. It would seem this is not analogous to a contradiction about some third matter, where perhaps we can maintain them by their presumption. Logically there is a difference, because in the case of contradiction about a third subject only one side is telling the truth; whereas in a head-on contradiction about the witnesses themselves it is possible that both sets are generally liars yet truth-tellers in this particular case.[6] From the Gemara’s and the Rishonim’s silence it seems they assume these two cases are identical.

In my lectures on the second chapter of Ketubot (Lecture 35) you can see further ramifications in the Trei u-Trei sugya that are also connected to our discussion here; I will not expand on this now.

[1] They are not propositions because they do not assert anything about the world. Consequently, they cannot be judged in terms of truth or falsehood. But sometimes judging in terms of truth or falsehood serves as an indication that helps us test whether some sentence is a proposition. This can be seen via logical implication: a sentence that asserts something about the world can be judged in terms of truth or falsehood (). That is the substantive direction. Negating the consequent gives us the indicatory direction: if it cannot be judged in terms of truth or falsehood, then it is not a proposition (). This relates to what I called in the fourth notebook “the theological argument,” and I will not expand on this.

[2] See also column 200.

[3] See my lectures on the second chapter of Ketubot.

[4] On admission by a litigant and its connection to be-yado, see column 306; for more on shavyei anafshei, see the shiur referenced in this question.

[5] One can quibble, since there are proofs that can nevertheless decide—if they do not merely join one set, but rather disqualify the other. Or the sets cancel one another and we are left with a decisive proof for one side. But I will not enter into all that here.

[6] Note that there is no contradiction between the contents of their testimonies. In column 195 I explained that the Liar Paradox cannot include a universal quantifier.