Causality: VII. Causality as an Internal Timeline (Column 466)

Contents of the Article

With God’s help

Causality: 7. Causality as an Internal Time Axis

Introduction

In the previous column I returned to the model described in column 33, according to which the time axis is composed of two components, static and dynamic. The static axis is the first component in the causal relation, that is, the cause must precede the effect on that axis, but not necessarily on the dynamic axis (at least when we are dealing with meta-legal questions. On the physical plane, it seems that it does).

In this column I would like to conclude this series, and I intend to address that internal time axis and its connection to the causal relation between events. I will do so through two discussions: one is R. Shimon Shkop’s proposal for stopping loops in Jewish law, and the second is the issue of “they take effect together.” At the end I will return to summarize our findings throughout the series and close the circle.

Stopping loops

In column 407 I presented R. Shimon Shkop’s proposal for stopping loops in Jewish law. Since the matter was discussed there, I will touch on it here only briefly, just to show the connection to the discussion conducted in this series about the relation between causality and time. The example that opened the discussion there was a case brought in the discussion in Gittin 83a, in which Reuven divorces his wife Leah on condition that she not marry Shimon. Leah married Levi and had children with him, divorced him, and then went and married Shimon (thereby violating Reuven’s condition). At first glance, a loop is created here that there is no way to stop. If she marries Shimon, her divorce from Reuven is voided, and therefore her marriage to Levi is invalid and her children by him are mamzerim, but now her marriage to Shimon is invalid as well, since she is Reuven’s wife. But if her marriage to Shimon is invalid, then she did not violate the condition, and therefore she is divorced from Reuven and her marriage to Levi is valid and her children by him are not mamzerim. But then her marriage to Shimon is valid, and once again the condition of the divorce from Reuven is undone, and so on endlessly.

R. Shimon Shkop, in his book Sha’arei Yosher (gate 7, ch. 16, p. 258), addresses this difficulty and writes as follows:

Because these marriages cannot take effect, for if they were to take effect they would be retroactively nullified by the condition. And likewise, anything whose taking effect is impossible does not take effect at all. Therefore the divorce stands and the marriages are void.

He proposes to stop this loop by means of a meta-legal rule that I called the “principle of consistency,” according to which a legal effect that would undermine itself cannot take effect (as a matter of principle, even without actual self-nullification). According to this, her marriage to Shimon cannot take effect, because its taking effect would circle back and uproot it, and therefore the loop stops here. She remains divorced from Reuven and properly married to Levi, her children by him are not mamzerim, and in the end she is also divorced from Levi, and that is that. Ultimately, she sits alone like a widow, bereft of all three husbands.

In that column I explained that this principle alone is not enough to stop the loop, and I showed that implicitly R. Shimon assumes another principle: a chain of events that occurs simultaneously is treated under Jewish law as sequential events that occur one after another along some virtual time axis. In other words, the causal connection between the events turns them, at least from the legal point of view, into non-simultaneous events, but into events occurring in a chain of cause and effect along an internal time axis of the occurrence. This is R. Shimon’s second assumption, according to which we must spread the events out along this internal axis.

In this picture, the legal view of the developments in the divorce example looks like this:

Her divorce from Reuven took effect when he divorced her. After that (on the actual time axis), her marriage to Levi also takes effect, and so does her divorce from him (again, this of course happens at a third moment, later on the time axis). Her marriage to Shimon comes afterward on the time axis, but it cannot take effect because its taking effect uproots itself (according to R. Shimon Shkop’s principle of consistency). The process of this self-nullification is a chain of events described along an internal time axis, as follows: her marriage to Shimon causes the nullification of her divorce from Reuven, and therefore her marriage to Levi is nullified and her divorce from him becomes irrelevant, and finally, by force of this chain, her marriage to Shimon is also nullified. Once we get here, we can see that her marriage to Shimon would nullify itself, and therefore it cannot take effect and this chain is never activated at all (this is a hypothetical calculation).

The nullification process is a virtual chain that takes place along an internal time axis of the case (on the actual time axis all of this happens at one moment). And once it is complete, it disappears. I should note that in that column I brought an example of a loop whose stoppage requires only R. Shimon’s second assumption (the spreading out of the events), without any need for the principle of consistency.

Notice that these processes include movement backward and forward along the real time axis. The marriage to Shimon that sets the chain in motion occurs at a later time, and it retroactively nullifies the divorce from Reuven that took place years earlier. That, in turn, nullifies the marriage to Levi that occurred after it, and so on. All of this happens simultaneously on the real time axis, but the processes within that moment include movement backward and forward along the time axis (this is the dynamic axis t defined there). According to R. Shimon’s second assumption, we take this entire loop on the dynamic time axis—which, as noted, does not actually occur at all (it is a hypothetical calculation)—and stretch it out so that everything is arranged along an internal time axis in a process that advances only forward along it (similar to the static axis in the previous column). What lays out this axis is the causal relation between the events, which in this case defines the time axis rather than relying on it. This is similar to what we saw in the previous column regarding the static axis, which also flows only forward according to the causal order (I remind you that we saw there that the temporal component of the causal relation refers to the static axis. On the dynamic axis, reversed causation may exist).

The conclusion is that, according to Jewish law, a causal connection between events in a chain creates an internal time axis that arranges them along it in forward progression from cause to result. This reflects the first component of the causal relation (the time component), but in a different order: the causal connection creates a “temporal” relation, rather than time being included as a component within the causal relation. We will now see this in another discussion.

Introduction: the connection to the issue of “they take effect together”

EA asked me here (this was a comment following the first column in the present series) how I would explain the rule of “they take effect together”:

How does the Talmudic rule that things “take effect together” work according to the model you proposed? It seems to me that it has the logical component (if you deliver the deed of manumission, the slave is freed), and also the physical component (the delivery causes the slave to be freed), but it lacks the temporal component.

He notes that in the rule of “they take effect together,” the temporal component is missing. At first glance he is right, but we will now see that it actually is present, except that it is the static axis, and here too it appears in an internal way. This is not the dynamic axis (the ordinary time axis).

The rule of “they take effect together”^[1]

The Mishnah in Gittin 77a states the following rule:

If one throws a bill of divorce to his wife and she is inside her house or inside her courtyard, she is divorced.

A man can deliver the bill of divorce to the woman by placing it in her courtyard, and this is considered as though she received it in her hand. The Gemara there asks:

Her courtyard? But whatever a woman acquires is acquired by her husband!

After all, her courtyard is acquired by her husband, and therefore, as long as she is married, he has no way to place the bill of divorce in the woman’s courtyard (as required for divorce), because she has no courtyard that is hers. In another formulation, the difficulty may be presented as follows: the woman cannot own the courtyard as long as she is married (because whatever a woman acquires is acquired by her husband), but if she has no courtyard, she has no way to stop being married (to receive a bill of divorce). The process cannot get started. If she can be divorced in some other way, then she can have a courtyard, and if she can acquire a courtyard in some way, then he can divorce her by placing the bill in her courtyard. But when she is married and wishes to be divorced by this very method, it is impossible. This is a loop in which each link depends on the other, and neither can occur without the other having preceded it. Seemingly, we are stuck.

Later in the Gemara (ibid., 77b), Rava explains it as follows:

Rava said: Is her hand any different? Does not her hand too belong to her husband? Rather, her bill of divorce and her hand come together; so too here, her bill of divorce and her courtyard come together.

Ravina said to Rav Ashi: Was Rava really troubled by a woman’s hand? Granted, the husband acquires the product of her labor, but does he acquire her hand itself? He said to him: Rava’s difficulty concerned the hand of a slave, according to the one who says that with a deed delivered through the slave himself, the hand of a slave is like the hand of his master. Rather, his deed and his hand come together; so too here, her bill of divorce and her courtyard come together.

In conclusion, the courtyard is acquired by the woman or the slave precisely at the moment of divorce or manumission, and therefore they can acquire their writ through it. The acquisition of the courtyard and the receipt of the writ occur simultaneously (as one).

The difficulty is obvious. After all, there is a dependency between acquiring the writ through the courtyard and the freeing of the woman. What help is there in the fact that they take effect together, if neither can take effect without the other? In other words, for the courtyard to acquire, the woman must first be divorced, and for her to be divorced, the courtyard must first acquire. The simultaneous occurrence of the two things should not help. It is as though we said both that the woman acquired the courtyard and afterward acquired the bill of divorce through it and was divorced, and that she was divorced and afterward acquired the bill of divorce in the courtyard.

Ketzot HaChoshen, sec. 200, subsec. 5, discusses this, and his claim is that the rule of “they take effect together” was said only לגבי a woman’s bill of divorce and a slave’s deed of manumission, where no act of acquisition is required, only delivery into their possession (in his view this would not work for transferring a courtyard by means of a deed, when I convey the deed to the buyer by placing it in the courtyard being transferred). According to his approach, the matter proceeds as follows: she receives the bill of divorce even though the courtyard does not belong to her, and at the moment of receiving the bill the divorce takes effect, and then the courtyard is her hand (even though it is not owned by her), and this happens simultaneously with the giving of the bill (apparently he holds that if no act of acquisition is needed, then there is no relation of cause and effect here, and therefore there is no obstacle to its happening simultaneously). The moment she is divorced, she of course regains her courtyard (for a woman’s melog property returns to her upon divorce). In the end, after she is divorced, she also acquires the bill of divorce through courtyard-acquisition (remember, in his view there is no need to acquire it as a condition of the divorce).

I should note that several later authorities raised objections to Ketzot HaChoshen and argued that acquisition of the bill of divorce is indeed required.^[2] Others argued that there is still a problem in the simultaneity of the creation of the hand (even if no act of acquisition is required), and that his proposal does not really change the situation in any essential way. Some interpreted Ketzot HaChoshen’s view (see, for example, Kehillot Yaakov Gittin sec. 29 and Imrei Moshe sec. 25 subsec. 6, in a somewhat different style) to mean that the process occurs in two stages: the delivery creates for her a hand (that is, a domain of her own), the hand thus created acquires the courtyard and the bill of divorce within it, and now that she has acquired the bill of divorce, she is divorced.

Some of the later authorities offered a different explanation of the rule of “they take effect together” (see Kehillot Yaakov on Gittin there, and similarly Achiezer, Orach Chayim sec. 28 subsec. 11, and somewhat differently Chazon Ish, Choshen Mishpat, Likkutim, end of sec. 8). In their view, in principle the woman does have a hand with which to acquire, but the husband’s hand blocks it. Delivery of the bill of divorce removes the obstruction of the husband’s hand, and now the woman can acquire.

And in Kehillot Yaakov, Gittin sec. 8, and in the novellae of R. Reuven on Kiddushin sec. 5, in a third approach, they wrote that it is incorrect to say that a slave and a woman lack the capacity to acquire. In principle they do have such capacity; rather, the master or husband acquires from them whatever they acquire. Therefore, in divorce or in the freeing of a slave, they really do acquire the writ, but at the moment they acquired it they were freed, and consequently the writ does not pass to the master or husband and remains with them as their property.

The difficulty in all the explanations

What all these explanations have in common is that there is not really simultaneity of two processes here. They all break the divorce process into two stages, and one happens after the other, in such a way that the dependencies are satisfied as required. True, this solves the logical problem, but it seems strained in the language of the Gemara. Remember that it resolves the difficulty by saying that they “take effect together.” Seemingly, that means that the two processes occur simultaneously, and that is what solves the problem. But this is of course very difficult, as I explained above, and therefore the later authorities strain the language of the Gemara in order to save the reasoning. Yet according to our approach above, it seems that we can nevertheless understand the Gemara literally.

Another explanation: causality as an internal time axis

In the book Or Gadol, sec. 53, it is written that the rule of “they take effect together” is a special innovation of the Torah. From the very fact that it said that a bill of divorce or manumission must be conveyed to a woman or a slave, it is clear that this difficulty does not trouble it. We see that it treats this process as two events that take effect together.

It seems that this can be understood in a way that fits the descriptions I presented above. We saw that sometimes Jewish law views a causal chain of events that occur at the same moment as if they were spread out along an internal time axis. If so, this itself may be the Torah’s innovation regarding the freeing of a slave and the divorce of a woman. The conveyance of the bill and the courtyard brings about the divorce, and the divorce brings about ownership of the courtyard, which in turn validates the acquisition of the bill and the divorce. We have here a causal circle of events that occur at one moment on the real time axis, but because there is a causal relation among them, they are regarded from our standpoint as spread out along an internal time axis (exactly like the loop of the nullification of the divorce, which goes backward in time and forward again with its tail in its mouth, and Jewish law “straightens” it out along an internal time axis determined by the causal relations among the events).

This can be formulated as I suggested above: the acquisition of the courtyard precedes the woman’s divorce and brings it about, and the woman’s divorce also precedes the acquisition of the courtyard and brings it about. In a circle, precisely as such, there is no first stage and second stage; rather, the tail of each stage is held in the mouth of the other. In column 195 I pointed to two logical structures, paradox and anti-paradox, and illustrated them by means of drawings by Escher. The picture of the anti-paradox is built of two hands drawing one another:

Here too there are two events, each dependent on the other and supposed to appear only after it: without the left hand—the lower one in the drawing—drawing the right hand—the upper one—the right hand does not exist. But the left hand is itself drawn by the right hand. The two events occur together on the ordinary time axis, and of course this cannot happen (because each is the cause of the other). But if, from the standpoint of Jewish law, there is an internal time axis wholly contained within a single moment of the ordinary time axis, then it can happen there. This is exactly what we saw above in the discussion of R. Shimon Shkop’s principle of consistency.

Note: must a cause necessarily precede its effect?

In our discussion here we assumed that the cause must appear before the effect (the first component of the causal relation. See column 459). But this relation is not simple. Ancient philosophers already noted that there is a logical argument leading to the conclusion that they should, on the contrary, appear simultaneously. Suppose that A is the cause of B. We saw that this means that if A obtains, then B necessarily obtains as well (because A is a sufficient condition for B. See column 462). But if A precedes B in time, then there is one moment in time at which A obtains and B does not, and that contradicts A’s being sufficient for B. On this basis, for example, they explain the relation between the fire and the light it emits (for purposes of the discussion we will deal only with the light that exists in the place of the fire itself, not with its illumination farther away). The fire is the cause of that light, but they exist simultaneously: when there is fire, there is necessarily light as well. There cannot be a moment in time at which there is fire and it is not emitting light.

In Milot HaHigayon by Maimonides, chapter 12, he wrote:^[3]

Of a thing that is prior to another, we speak in one of five ways. The first is priority in time, etc.; the second is priority by nature, etc.; the third is priority in rank, etc.; the fourth is priority in excellence, etc.; the fifth is priority as cause, namely: when two things are equal and coextensive in existence, and neither is found without the other, except that one of them is the cause of the existence of the other. Then we say of the cause that it is prior to the effect, just as we say that the rising of the sun is prior to the light of day, even though they are together, because the rising of the sun is the cause of the existence of the day.

At first glance, the priority of the cause to the effect is only a metaphor, since on the time axis they appear together. I will qualify this below.

Take as another example Newton’s second law of mechanics: F=ma (force equals mass times acceleration). We saw that it is customary to interpret this equation in such a way that the force is the cause of the acceleration and not the other way around (although the equation itself contains no hint of this). Moreover, the equation is simultaneous, meaning that if the force changes with time, the acceleration changes accordingly, and at every moment t it holds that the force at that moment equals the mass times the acceleration at that same moment. If so, the cause (the force) and the result (the acceleration) appear at the very same moment. The force does not precede the acceleration. So too, a change in the force appears simultaneously with the change in acceleration caused by it. How is all this reconciled with the accepted assumption that the cause must appear before the effect?

Before I address that, notice that the simultaneity of cause and effect also raises a difficulty from the opposite direction. How can there be a causal chain that extends over some duration? If every effect appears at the same moment as its cause, then the next effect in the series will also appear at that same moment, and so will the whole chain of subsequent effects. But we know that there are causal processes that extend over some duration on the time axis (see about causal chains from another angle in column 462). How can this be?

In field theory in physics, however, these processes are indeed described in a way more suited to reality. There, the force produces acceleration in a gradual process that takes time. Just as one mass sends out a force that will draw another mass, but it takes time for that force to reach and act on the second mass. There one speaks not of gravitational force but of a gravitational field that advances through space and time.

The meaning is that the more accurate description of the causal relation is indeed not simultaneous. It is a continuous process that takes place along the time axis from moment to moment, and this may be seen as a continuous chain of causes over the time-points on the axis. Duration is a parallel result of the formation of segments from points on the continuum (temporal or spatial). In this description, the cause does indeed precede the effect, but since this is a continuous process, it spreads out over the real axis, and therefore one cannot speak of a moment at which a cause or an effect occurs (see a similar discussion of Zeno’s arrow paradox in my article here and in column 170), and consequently one cannot speak of a moment at which the cause exists and the effect does not. They touch one another like points on the real axis. This confusion is very similar to the fallacy that underlies all of Zeno’s paradoxes of the continuum.

Closing the circle

It seems to me that this description places in a somewhat different light the claim about the temporal priority of the cause to the effect. And the very notion of an internal time axis, all of whose events occur one after another yet all take place at a single time-point on the real time axis, suddenly appears a bit more intelligible.

If you look again at the quotation I brought above from Milot HaHigayon by Maimonides, you will see that he speaks of the cause preceding the effect even though on the time axis they appear together. At first glance this is only a metaphor of priority in the causal sense, not the temporal one. But in truth there is a sense that one does precede the other, even though cause and effect appear together at the same moment of time (like the shadow together with the body that casts it, or the fire and the light it creates). So whence comes this sense of priority? Why do we use the expression ‘before,’ which belongs to the time axis, also with respect to cause and effect? In light of what I have said here, it seems that this is not merely a metaphor, but that there really is a priority of cause over effect along the internal time axis, even if they are simultaneous on the ordinary axis.

The end bound to the beginning: a summary of the whole argument

In the opening column we saw three components of the causal connection: the temporal, the logical, and the physical. After the whole course of this series, in order to close the circle we will return to these three characteristics and summarize what we have brought to light regarding each of them.

Suppose that event A occurs at time (TA,tA)) and event B occurs at time (TB,tB). If we say that there is a causal relation between them, that is, that A(TA,tA) is the cause of B(TB,tB), this means that three requirements are satisfied:

1. The temporal component: TB>TA.

But not necessarily tB>tA, at least when we are dealing with abstract, nonphysical axes. On the physical plane it would seem that tB>tA is required as well.

1. The logical component: B(TB,tB) 🡪 A(TA,tA).

This is a material logical implication. A is a sufficient condition for B, that is, it cannot be that A obtains and B does not obtain, or that if B obtained then A necessarily obtained as well.

1. The physical component (causation): A(TA,tA) 🡺 B(TB,tB).

Here the arrow is not a material implication but a symbol for physical causation: A brings about B.^[4]

If one of these components is missing, then we are not dealing with a causal relation. For example, if the influence is backward in time t (and not T), that is not causality. If A is a necessary but not sufficient condition for B, that is not causality. And if the physical component is missing, then one may say that there is a correlation, but not a causal relation.

1.

Footnotes

1. For a brief discussion of this, see my written lectures on Gittin, Lesson 24 (and the lesson before it).
2. See, for example, Even HaEzel, Hil. Hiring 9:3; the novellae of R. Shimon Shkop on Gittin, sec. 5; Kehillot Yaakov on Gittin, sec. 8; Kovetz Shiurim on Kiddushin, sec. 138; and more.
3. See also Guide of the Perplexed, Part II, chs. 20–21, and Mefa’ane’ach Tzefunot by R. M. M. Kasher, at the beginning of chapter 10, among others.
4. As was rightly noted in a comment on the first column, this component includes them all. That is, if it obtains, then both the temporal component and the logical component obtain as well, but it contains something beyond them. Incidentally, if you go through the comments on that column, you will find that all of them were answered later in the series.