Causality: A. A General Overview (Column 459)

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.

Past experience shows the great popularity of series of columns (the number of comments starts in the single digits on the first column and quickly plummets. Toward the end it turns negative and finally complex). This is a good reason to launch another series, this time on causality and its branches. I have dealt with this topic in several places (mainly in Chapter Six of my book Science of Freedom and in the opening part of the fourth book in the Talmudic Logic series), but I thought it appropriate to lay out it and its relevant contexts in a focused way. I will begin the discussion with the matter of causality and adjacent issues, and then continue to backward time and praying for what has already occurred (these topics, too, have been discussed here in the past).

The three components of causality

There is a common assumption called the “principle of causality,” according to which every event in the world is supposed to have a cause; or, in the inverse formulation: nothing happens without a cause. Things do not just happen on their own. Consequently, when we examine any event, we will always look for its cause. When we see a ball flying, we will look for who or what kicked it. If we see a charred area, we will assume there was a fire. If we have an illness, we will look for its cause, and so on. As a rule, we are unwilling to consider the possibility that something happened without a cause. The question we will address here is the nature of the causal relation. To sketch the framework of the discussion, I will actually start from the end.

The definition I proposed in the aforementioned sources for the causal relation consists of three components: the temporal, the logical, and the physical. Each of the three is contested, but as I will try to show below, when one drops any single one of them, one may arrive at analytic mistakes and erroneous conclusions. For the opening survey, I will denote the cause-event as A and the effect as B. To claim that there is a causal relation between A and B, we must ensure the existence of three elements that constitute this relation:

• The temporal component. The cause A must occur before the effect B along the timeline. That is, for A to be considered the cause of B, it must hold that: t(A) < t(B) (where t(x) denotes the time of event x).

The dispute regarding this component revolves around whether reverse causal influence is possible—namely, a causal relation in which the cause appears after the effect. This can be asked on the conceptual plane (can such a causal relation be defined) or on the practical plane (even if in principle it can be defined, do there actually exist phenomena of reverse causal influence).

But temporal precedence alone is not sufficient to define a causal relation. No one would say that the sunrise on Tuesday is the cause of the fall of the government that occurred on Thursday afterward, since there is no logical conditioning between these two events, only temporal precedence. Therefore, we must add two further elements to the definition of the causal relation.

• The logical component. The cause A must constitute a logical condition for the existence of the effect B. That is, for A to be considered the cause of B, it must hold that: A -> B (the arrow symbolizes logical entailment). One could, of course, say that Tuesday’s sunrise is a logical condition for the fall of the government, but that is mere formalism. Clearly, the fall of a government can occur on any day; therefore, this condition is not substantive and is not truly necessary for the occurrence of the effect (B). The timing of the fall is accidental. The convening of the Knesset is already a substantial logical condition for the fall of the government.

The debate regarding the logical component revolves around whether it suffices that the cause A be a sufficient condition for the effect B, or whether it must be a necessary and sufficient condition (a merely necessary condition, everyone agrees, cannot be considered a cause).[1] I will define and clarify this further below.

But even temporal precedence and a logical relation are not enough to fully define the causal relation. The relation between Tuesday and Wednesday satisfies both the temporal condition and the logical relation. If today is Tuesday, then necessarily tomorrow will be Wednesday (here we are dealing with a necessary and sufficient condition, so the logical requirement is satisfied in exemplary fashion and to everyone’s satisfaction), and yet I would not say that the relation between the days is causal, i.e., that Tuesday is the cause of Wednesday. This is also a substantive condition, since the relation between the days is certainly not accidental (unlike the relation between the sunrise and the fall of the government). What is lacking here is the physical relation—production. Tuesday indeed always precedes Wednesday, but one cannot say that it is the producer of the fact that tomorrow will be Wednesday. Here we arrive at the third requirement for the causal relation.

• The physical component. The cause A must be the factor that produces the effect B. We can denote this with a double entailment arrow (a single-line arrow denotes logical entailment): A => B. The kick I give the ball is not only temporally prior to its flight, nor is it merely a logical condition for its flight; it is also that which produces its flight. Here all three requirements are met, and therefore one can say that the kick is the cause of the ball’s flight.

This component, too, is contested, chiefly by David Hume’s argument denying the existence of the physical component. He claims that the definition of the causal relation cannot contain such a component (to be explained below).

This is only a general survey to clarify the framework of the discussion. Later I will delve into each of these three components in greater detail, but first I will clarify the relation between the components using a halakhic example.

Example: Beginning in negligence and ending in force majeure

In Column 248 I briefly addressed this example, and here I will spell it out a bit more. In several places in the Talmud there is reference to a situation in which a person enters, through negligence, into a state of force majeure (ones). The main Talmudic discussion of this concerns the laws of bailees (shomrim). By way of preface: when a person accepts an object for safekeeping, he must take care to guard it properly. If he does not, he endangers the object, as it may be damaged or stolen, and then he will have to compensate the owner. However, if he guarded it properly and nonetheless something happened to the object, the law is that the bailee is exempt, since he was under compulsion (an unavoidable mishap).

Now consider a person who receives a wallet with banknotes for safekeeping. It is known that if he places it in a forest it may burn, but there is no concern that it will be stolen (thieves do not search for wallets in a forest). By contrast, if he leaves it in an unlocked house it may be stolen but will not burn (let us assume this, at least for the sake of discussion). Now consider a case where the bailee places the wallet in a forest, and indeed a fire breaks out there. In such a case he was negligent in his guarding, and therefore must compensate the depositor and pay him the money. The same applies if he leaves the wallet in an open house and it is stolen. But what if he left the wallet in a locked house and it was stolen? Clearly he is exempt, because theft from a locked house is force majeure (ones). And what if the house burned down? That too is force majeure, because a house is not expected to burn (whether open or locked), and therefore he is exempt. But what is the law if he leaves the money in an open house—which is negligence due to the risk of theft—yet in the end there was no theft but rather a fire? A similar question arises when he hides the wallet in the forest (which is negligence with respect to fire), but in practice no fire breaks out and instead a theft occurs. These last two cases are called in halakhic jargon “beginning in negligence and ending in force majeure.” The situation begins with negligence due to a concern for event X, but ultimately an unforeseen event Y occurs, which in this situation is defined as force majeure. There was negligence here, but in the end force majeure occurred.

In Bava Metzia 42a there is a discussion of a case in which a bailee received coins as a deposit, hid them in a shack in a forest (he was negligent vis-à-vis fire but not vis-à-vis theft), and the coins were stolen. This is precisely a case of “beginning in negligence and ending in force majeure,” and the Gemara there brings a dispute about whether he is liable or exempt. The halakhic ruling in such a case is that the bailee must pay; in the Talmud’s language: “Beginning in negligence and ending in force majeure—liable.”

Tosafot there write that this is the law only if, without the negligence, the force majeure would not have occurred. In the example cited by the Gemara there, had the bailee not been negligent and hidden the wallet in the forest, it would not have been stolen, and therefore he is considered responsible for the theft and must pay. Granted, one cannot say that hiding it in the forest caused the theft (since money hidden in a forest is not expected to be stolen), but it is still correct to say that without that action the theft would not have happened. By contrast, in cases where there is no connection between the negligence and the force majeure (the mishap would have occurred regardless of the negligence), the bailee is exempt from payment. Such a case is found in Bava Metzia 36b, where the bailee was negligent with an animal under his responsibility and did not lock it properly in the cowshed (this is negligence with respect to theft or the animal’s escape), and it went out to a pond, where it died in the ordinary course of nature (not because it bumped into something or was harmed along the way, but simply died as is the way of all flesh—there it is called “the angel of death”). The assumption is that the animal would have died even had it remained in the cowshed; its time simply arrived. In such a case, the halakhic ruling is that the bailee is exempt from paying the depositor, since in any case—even had it remained in the cowshed—it would have died. In the Talmud’s language: “What does it matter to the angel of death if here or there.”

Note that had the animal encountered an obstacle in the pond and died because of that, all would agree that he would be liable. That is a case in which there is a causal connection between the negligence and the force majeure. Therefore, legally, the negligence is considered what caused the damage, since that is what was to be expected. If that had been the situation, then there would be no force majeure here at all, and it would not be defined as a case of “beginning in negligence and ending in force majeure,” but rather as simple negligence. A more extreme case is one in which the animal is stolen after he failed to lock the door properly. That is certainly a case of outright negligence (“kulah peshi’ah hi”—“it is all negligence,” in the Gemara’s language), and all agree he is liable.

Let us conclude with an additional novelty. The Rif’s position (on the sugyah in Bava Metzia 36) is that according to one amora (Abaye), the bailee is liable even in a case where there is no connection between the negligence and the force majeure (when it died in the ordinary course of nature. In his view we do not say the reasoning of “what does it matter to the angel of death if here or there”). Granted, the halakhah does not follow that opinion, but what matters for our purposes is that such a view exists in the Talmud as well.[2]

In summary, we have presented three types of cases:

• The bailee was negligent and hid the wallet in the forest, and a fire broke out. This is outright negligence (since forests are prone to fires), and by all accounts the bailee must pay.
• The bailee was negligent and hid the money in the forest, and thieves came. This is a situation of beginning in negligence (regarding fires) and ending in force majeure (theft, which is not expected for money hidden in a forest), and there is a dispute about it. The halakhic ruling is that the bailee is liable even in this case.
• The bailee was negligent and left the door open, and the animal went out to the pond and died there in the ordinary course of nature (it would have died even had it not gone out). In this case, there is indeed an exceptional opinion that the bailee is liable, but the halakhah is agreed by all that he is exempt.

My claim is that these three cases/situations express the three components of causality we have discussed so far. To see this, let us examine the relation between the bailee’s actions and the damage that occurred afterward in each of the three cases.

Case C is a situation in which there is no connection between the negligence (leaving the door open) and the force majeure (the animal’s dying in the ordinary course of nature). The only one of the three components that characterizes the causal relation here is the temporal component: the animal’s death occurred after the bailee’s negligence. The logical relation does not exist here, since a logical relation is always of the form “if… then…”. Here it is not correct to say, “If the bailee opens the door, then the animal dies” (it would have died even without that).[3] All the more so there is no physical relation (a relation of production) between them, since it is not correct here to say, “The animal died because the bailee opened the door.”

What happens in Case B? Here, too, there is of course the temporal component, since the theft occurred after the wallet was hidden in the forest. The logical component is also present, since in this case it is correct to say, “If the bailee had not hidden the wallet in the forest, it would not have been stolen.”[4] But the physical relation does not exist here, since it is not correct to say, “The wallet was stolen because of the negligence.” In the legal sense (as distinct from the physical sense), there is no relation of production here, since, as we saw above, with respect to theft, hiding it in the forest constitutes proper guarding; therefore, there is no legal responsibility of the bailee for what happened.

What happens in Case A? Here, too, there appears to be the temporal relation (for it is correct to say, “The burning of the wallet occurred after it was hidden in the forest”). The logical relation is also present (for it is correct to say, “If the bailee had not hidden the wallet in the forest, it would not have burned”), and here the physical relation exists as well (for it is correct to say, “The burning of the money occurred because of the hiding in the forest”), with the “physical” production replaced here, of course, by “legal” production (at the legal level the hiding in the forest is the factor that led to the incineration of the wallet).

Thus, the Talmudic discussion of “beginning in negligence and ending in force majeure” precisely distinguishes among the three components of causality as we have presented them here: if there is a full causal relation (all three requirements), there is a duty for the bailee to pay. In the situation of beginning in negligence and ending in force majeure, only the logical and temporal components exist, without the physical (legal) component, and then there is a dispute. The halakhic ruling is that the bailee is liable even in such a case. And if only the temporal relation exists, this is not causality at all, and therefore the bailee is exempt (except according to Abaye as per the Rif).[5]

I will now move on to examine the connections among the three components of the causal relation and the disputes about them, beginning with the relation between the first and second components.

The relation between the temporal component and the logical component: logical determinism

In Column 301 I described an argument called “logical determinism,” and here I will briefly draw on it. It is an argument for determinism based on the following logical consideration: when I assert today, “There will be a sea battle tomorrow,” I have no way today to know whether the statement is true or not. Suppose that tomorrow a sea battle does indeed occur; it then turns out that the statement is true. But in that case it was true from time immemorial (we just did not know it). The reason is that the truth of a statement is grounded in a comparison between its content and the state of affairs in the world that it describes. If a sea battle did indeed occur on Wednesday, then the statement “On Wednesday there will be a sea battle” corresponds to the state of affairs in the world that it describes and is therefore true. This is, of course, correct at any time whatsoever (even on Monday). The fact that we cannot know the truth value of the statement on Monday does not mean it has no truth value at that time.

Therefore, the statement “There will be a sea battle tomorrow” is already true today. But if already on Monday the statement “On Wednesday there will be a sea battle” is true, then on Wednesday a sea battle must occur (otherwise it would turn out that the statement is not true and never was). The conclusion is that if on Wednesday a sea battle occurred, nothing else could have happened. What will occur there is fixed from the six days of Creation. Even if no sea battle occurred on Wednesday, then for the very same reasons it is clear that the statement was always false. This is true, of course, for every event under the sun, whether or not anyone uttered a claim about it. From here an apparently deterministic picture emerges: every event that happened or did not happen at some time had to happen (or not), i.e., it could not have failed to happen. Everything that happens has been foreordained since the six days of Creation, and free choice is not granted.

What will the libertarian (one who believes in free will) answer? The answer I proposed there was that the truth value of a statement is not a kind of fact, or a claim about reality. It is a logical definition, a product of our own thought, and nothing more. Therefore there is no impediment to a future event’s determining the truth value of a statement in the present or the past. A future event cannot be the cause of an earlier event (this is the temporal component of the causal relation), but the truth value of a statement is not an event and therefore cannot be the cause of anything, and the future event is not a cause that produces it, since what is not an event does not “occur.” Therefore there is no necessity that the event occur before the truth value of the statement that describes it is fixed.

In other words, the relation between the truth value of a statement and its content is not a causal relation. The one is indeed defined on the basis of the other, but it is not produced by the other. In general, definitions and logical relations are not subject to the timeline, and therefore there is no impediment to the truth value of a statement in the present (which is, of course, its truth value at any time) being determined by a future event. Thus there are indeed two possibilities regarding what will occur on Wednesday: a sea battle may occur, which will retroactively fix the statement’s truth value as “true” since the Creation, or it may not occur, which will retroactively fix that the statement was always false. The truth value of the statement at some time is not a fixed, unchangeable fact but a definition that will be filled with content after the event occurs, and there is no impediment to this happening retroactively.

This is a good demonstration of the independence of logic from time. Logic is atemporal, i.e., indifferent to the timeline. Truth values of statements, like logical relations in general, have nothing to do with time. Hence, a logical conditional can certainly contain an antecedent that speaks about the future and a consequent that deals with the present or the past. Logic will have no problem with such a sentence. For example, if a person were to say: “If it rains tomorrow, then there were winds today,” there is nothing defective in this sentence (it can be true or false, but the order of times between the antecedent and consequent is not relevant to that issue).

The conclusion is that the first component of the causal relation, temporal order, is independent of the second component, the logical. The two initial requirements that A and B must satisfy so that we define the relation between them as causal—(a) t(A) < t(B); (b) A -> B—are independent. It should be noted that the third component, physical production, is not indifferent to time. The requirement A => B, at least according to the common interpretation (which does not allow reverse causality in time), implicitly presupposes the first relation (temporal precedence). And yet these are two different requirements, because the converse is certainly not true. In other words, the first requirement comes to say that physical production cannot occur backward in time (it comes to exclude the view of reverse causality).

We now turn to the relation between the third component and the first two.

The relation between physical production and the first components (time and logic): David Hume on causality

I have often mentioned David Hume’s claims about causality. The background to his arguments is his empiricist approach, i.e., his assumption that claims about the world can arise only from observation and not from other sources. Empiricism rejects the rationalist approach and argues that if something seems to us reasonable and logical, that does not mean that this is how the world really operates. Against this background, Hume argues that the causal relation is to be defined on the basis of the first two components, without the third. His rationale is that the third component has no observational source—that is, we cannot apprehend it empirically—and therefore we should not accept it. That causality seems logical to us does not mean that this is indeed how things happen in the world. It is a cognitive structure with which we were born, but there is no justification to impose it on the world’s actual conduct. I will explain this a bit more.

Hume’s claim attacks causality on two different levels. First, he argues that the principle of causality has no observational source. How can we know that every event has a cause? But this is a question that can be asked about every general scientific law. Even if we have seen a mode of behavior of nature in certain cases, how can we know that this law is truly general (that this is always what happens)? In the laws of science we rely on the assumption of induction, which, in a somewhat simplistic formulation, says that if something happens in various cases, that is probably what always happens. Granted, Hume also attacks the principle of induction in general, but with respect to the principle of causality the problem is more severe. Beyond generalization, even its appearance in any given case we have observed cannot actually be learned from observation. When we see someone kicking a ball and then the ball flying, we can, of course, see that the ball flies after the kick (the temporal component), and that if one kicks then the ball flies (this is always what happens after a kick—this is a sufficient condition, and therefore the logical component holds, and we arrive at it through generalization—induction). But how can we learn from this that the kick was the factor that produced the ball’s flight (the physical component)? That we cannot see with our eyes or by any other sense, and therefore Hume, as an empiricist, claims that this assertion is inadmissible. As noted, this is how our thinking works, but there is no reason (!) to assume that this is what happens in the world itself.

In light of the picture I have described above, we can say that Hume does not undermine the causal relation but only its physical component. He accepts the notion of cause and the causal relation but claims that they include only the temporal and logical components, not the physical component. And yet it is important to note that the very fact that he treats this as an attack on the principle of causality shows that he himself understood that the accepted definition of the causal relation also includes the physical component. His own intuition indicated as much, but, in his view, this is an illusion born of a rationalist approach (that what seems to us logical and reasonable is likely also factually correct).

Ironically, it is modern physics that has adopted Hume’s approach, and the accepted definition there of the causal relation does not include the physical component (production), but only the temporal relation and the logical relation. It is easy to miss the fact that equations of physics, sometimes taken as descriptions of cause and effect, are not actually such. Consider, for example, the relation between force (F) and acceleration (a), as determined by Newton’s second law of mechanics: F = ma. Intuitively it is clear to us that force is the cause and acceleration is the effect (a body accelerates because a force acts upon it). But this cannot be inferred from the equation itself. It only determines a relation between force and acceleration, but the two have completely symmetric standing. From the equation one cannot learn which is the cause and which is the effect. So it is with all the equations of physics, since equality always determines a symmetric relation.

Moreover, physics has no formal way at all to express entailment between two quantities (entailment is a non-symmetric relation: the antecedent entails the consequent). The fundamental relation in physics is equality, which, as noted, is symmetric in its essence. Thus, surprisingly, it is physics that drops the physical component from the definition of causality. In contemporary physics, when one speaks of a “causal relation,” the intent is mainly to a temporal relation and also to a logical relation. Production plays no role here.[6] Physicists, of course, speak about relations of production. Any physicist you ask will tell you that force causes acceleration and not vice versa, but that is our interpretation of the equations. They themselves are entirely symmetric, and they contain no distinction between force and acceleration. Therefore, empiricists argue that the causal interpretation we give things is only a mode of description and thought convenient to us because of the structure of our thinking, but it is not a claim about the world’s conduct itself.

And yet we all continue to speak about causality in its pre-Humean sense. Every physicist will tell you that force causes acceleration, and the term “causal relation” in its “thin” meaning (only time and logic) is perceived by us as pseudo-causality (causality in a metaphorical sense). As noted, even Hume himself assumed that this is the intuitive conception of the causal relation. Several arguments can be mounted against Hume’s position, but this is not the place. In brief, I will mention here my claim that empiricism assumes a mistaken premise. Our intuition is a cognitive instrument, and therefore an intuitive discernment of the existence of a causal relation (with its three components) suffices to validate it. Of course, this is not certain (nothing in the world is certain), but the burden of proof is on whoever claims that the principle of causality is not correct (or that the third component should not be included in it). In any case, for our purposes here I will continue to assume that a causal relation should contain all three components.

So far we have dealt with the necessity of the third, physical component. I will now return to a similar look at the first two components—the temporal and the logical.

Between correlation and causation

Statisticians distinguish between correlation and causation, and even tend to poke fun at laypeople who fail to make this distinction. A common joke says that dieting is inadvisable, since everyone who goes on a diet is fat. The claim that those who go on diets are fat is true (for the sake of discussion), but the conclusion is of course false. There is a correlation between fatness and dieting, but its direction is not from dieting to fatness but from fatness to dieting. People diet because they are fat, and they do not become fat because they diet. Granted, in this example there is a causal relation between the two variables and not just a correlation, but the causal relation is opposite in direction to what is claimed. This already teaches us that one must be cautious when inferring from the existence of a correlation to a conclusion about a causal relation.

But sometimes correlation does not indicate at all the existence of a causal relation between the two variables (in any direction), and this can appear in three forms: either it is accidental, or it stems from a relation that is not causal, or the causal relation is connected to a third factor. Accidental correlation is the most basic error. Every time I saw a horse, I saw a cart after it. Does that mean the horse is the cause of the cart’s appearance? Not necessarily. There are horses that go without carts and carts that move without horses. Here there is not even a correlation, since this is pure coincidence.

But in principle, there can also be a situation where two events always appear one after the other—i.e., there is correlation between them—and yet the relation between them is accidental. For example, even if always after a solar eclipse we saw that the third donkey from the left in Uncle Moshe’s farm scratched its ear, it could, in principle, be mere coincidence. Note: I said that here it happens always (unlike the horse-and-cart example), and yet it is possible that there is no causal relation here. It is the result of chance alone. In such a case, it is correct to say there is correlation, and yet there is no causation. This sharpens, of course, the meaning of the third component (production) and the problematic nature of Hume’s definition, which ignores it.

But even when there is necessary correlation, one cannot always infer from it to causation. Thus, for example, when I see that always after receiving a COVID-19 vaccine one goes to sleep in the evening, this does not indicate any connection between the events. Every day in the evening we go to sleep. Here there is correlation, and it is not even accidental (one can say this is always what will happen in the future as well, unlike the solar-eclipse example), but it is not correct to say there is a causal relation here. The same holds for Wednesday, which always comes after Tuesday. In this case there is correlation, but no causation (note that according to Hume’s definition, in all these examples there is a causal relation between the events, since in his view the causal relation does not include production).[7]

The third and final form of non-causal correlation is a correlation that stems from a causal relation between both phenomena and a third factor. I will illustrate this through the argument of the Jewish-American logician Raymond Smullyan, mentioned here several times in the past, regarding astrology. Astrology assumes that the state of the stars affects what is expected to occur here on our planet. A common argument against this thesis is based on the physical principle that no information can travel faster than the speed of light. If so, the present state of the stars might affect what will happen here many years hence, but one cannot infer from it conclusions about what will happen here in the coming days.

Although I am not a fan of astrology, for the sake of fairness it must be noted that this argument is, of course, mistaken. The “present” state of the stars is the appearance visible to our eyes today, but precisely because of the speed-of-light limitation, its source is the actual state of the stars many years ago. If so, what I now see is the state of the stars years earlier, and there is no impediment to its affecting events occurring here now (this influence takes place in parallel with the movement of the visual information about the stars—from their past state to our eyes today). But what is relevant for our purposes is the solution Smullyan offers to this puzzle.

His claim is that although the present state of the stars cannot affect what is happening in the world now, there may nonetheless be a correlation between the state of the stars and what is happening here, and the correlation arises due to the influence of a third factor. There is some third factor that affects both the stars and us in parallel and creates this correlation. Thus, even without a causal relation between the stars and what happens here (in either direction), one can still draw conclusions from the stars’ state about what will happen here. This is an excellent example of a non-causal correlation between A and B that is created due to a causal relation between both and a third factor C. Hence it is important to examine whether we are dealing with correlation or causation before drawing conclusions.

The philosopher Gottfried Wilhelm Leibniz, in his essay on monadology, offered another example of such a correlation. Consider two clocks, A and B, that at every given moment indicate precisely the same time. We have three interpretive possibilities for this situation (if we ignore the possibility that it is coincidental): (1) Clock A determines the state of clock B. (2) Clock B determines the state of clock A. (3) There is a third factor (the watchmaker) who set the states of the two clocks and ensured their synchronization. The correct interpretation is of course the third, and this is another example of a correlation that contains no causation.

In all these cases, what interests statisticians is only the question of whether there is a correlation—real or illusory. On the assumption that the correlation is real and always holds, that is what they call causation (in this sense they adopt Hume’s definition). Statisticians do not deal with production in the physical sense, because what interests the statistician is predicting the future in light of the data at hand, and for that purpose it suffices to understand that there is a real correlation even without production (time and logic alone). Therefore, in the example of Tuesday and Wednesday, from the statistician’s perspective, this is causation and not correlation, since if it is now Tuesday one can predict with certainty that tomorrow Wednesday will arrive. But from the ordinary perspective it is clear that this is only correlation and not causation (because the third component—physical production—is missing).

[1] Two notes for the cognoscenti:

• The standard entailment in logic (material implication) means that A is a sufficient condition for B. The logical notation for a necessary and sufficient condition is equivalence. We will touch on this below. The entailment symbol used above does not necessarily express the material sense, but a logical relation between cause and effect, whatever its character may be.
• In addition, it is important to note that the entailment notation appearing here is not formal; that is, it refers also to the contents of events A and B, and not only to the truth values of the propositions A and B. See on this in the thread here.

[2] See Tosafot, s.v. “Huchmah,” Bava Metzia 78a, who disputes the Rif. In their view even Abaye does not hold so.

[3] For the cognoscenti: I remind that the entailment here is not material.

[4] In the simple physical sense, this is indeed a necessary and not a sufficient condition, and as we have seen, such a logical linkage certainly does not suffice to create a causal relation. But even when the money in the forest would have burned, sufficiency is not satisfied. Therefore it is clear that at the level of legal liability sufficiency is taken as a given, and what remains is only the discussion about necessity. The mere fact that I accepted an object for safekeeping imposes upon me responsibility as if my actions were sufficient for the result, and the only question is whether they are also necessary. Therefore, in the discussion here, the logical relation is expressed only in necessity and not in sufficiency.

[5] Let us briefly complete the picture for the curious reader. Why indeed does the Rif think it is possible to hold liable even when there is no causal connection (but only a temporal one) between the bailee’s actions and the outcome? It would seem that, in his view, the bailee’s very negligence obligates him to pay, as if he himself stole or made the animal disappear. However, if he was negligent and in the end nothing happened to the animal, he “pays” by returning the deposited animal itself; therefore de facto he does not have to pay. But if the animal died—even if it died in the ordinary course of nature (unrelated to his negligence)—there still remains on him an obligation to pay; in such a case he obviously cannot return the animal itself (since it is dead), and therefore he must pay money in its stead. As noted, this opinion was not accepted as halakhah. In any case, according to this view the bailee’s liability does not require a causal relation between his actions and the outcome, and therefore it does not pertain to our discussion.

[6] The law of gravitation sets an equality between the force of gravity and the values of the masses that cause it and the distance between them. The equality is simultaneous (the force at moment t equals the product of the masses divided by the square of the distance between them at that same moment). To move to a relation of production one moves to a description called “field theory,” where one describes the propagation of the force from one mass to another over time. But even there one speaks of a force field, and it too is described by equations. At most the temporal dimension enters there, but certainly not the physical one (production).

[7] At least regarding COVID and sleep one can, of course, tweak the definition and demand an exclusive causal relation—that the one not occur without the other (this takes us into the discussion of sufficiency and necessity in the causal relation, which will be discussed below). But with respect to the relation between Tuesday and Wednesday, I think that according to Hume one cannot avoid treating them as cause and effect. This sharpens the absurdity of his approach.