Causality: C. A Look at the Logical Component of Causality (Column 462)

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.

In the two previous columns I argued that the causal relation has three components: temporal, logical, and physical, and I noted the difficulty of handling the physical component in a formal way (symbolizing it). I now wish to enter a more precise analysis of the logical component—and a bit of the temporal one as well. I will begin with a logical distinction between different kinds of conditions.

Necessary and sufficient conditions

First, let us mark the conditional in the logical sense (not necessarily the material sense; this is only logical notation for our purposes) in the form of an implication: A → B. Note that this symbol corresponds to the double/thick arrow I used in Column 459. The antecedent of the conditional (the “if”-part) is A, and the consequent (the “then”-part) is B. Logicians distinguish between two kinds of conditions: a necessary condition and a sufficient condition. These are two different meanings of the arrow → appearing in the relation above.

When we say that A is a necessary condition for B, we mean that B cannot occur unless A occurs. In other words, the occurrence of B implies the occurrence of A. Note that the occurrence of A can be earlier than B or later than it, since— as we saw in the first column—logic is indifferent to the time axis (this is another reason not to identify it with causality). We can, with equal justification, say that clouds are a necessary condition for rain (there can be no rain without clouds), and also that wet ground is a necessary condition for rain (it cannot be that rain fell and the ground did not become wet). In the second case, it is clear we are not talking about a cause, since the wetness of the ground arrives later than the rain (it is the result of the rain, not its cause). A cause must precede its effect, but a necessary condition need not precede what it conditions.

What characterizes both examples is that the antecedent does not entail the consequent. That is, the presence of clouds does not necessarily entail rain (if they are not rain clouds, or circumstances block rain), and wet ground does not necessarily entail rain (perhaps sprinklers were on).

Now we define another meaning of the arrow: a sufficient condition. When we say that A is a sufficient condition for B, we mean that if A occurs, then B will occur. In such a case, it is enough to know that A occurs to conclude that B will occur.

If we examine the two examples above, we will easily see that B is necessary for A whenever A is sufficient for B. For example, knowing that rain fell suffices to conclude that there were clouds (earlier), and equally, knowing that rain fell suffices to conclude that the ground became wet (afterward). Thus rain is a sufficient condition for both of these outcomes.

More generally, we can prove this by contradiction. Suppose that A is necessary for B. To prove that B is sufficient for A, assume the opposite—that B occurs but A does not. But since A is necessary for B, B cannot occur unless A also occurs. Therefore it cannot be that B occurs without A. The converse proof is identical: if A is sufficient for B, then B is necessary for A. Q.E.D.

A note on material implication

The logical relation A → B is not the causal arrow I marked with a double/thick arrow; it is a logical implication that can be defined in several ways. The common way is material (truth-functional). According to this definition, its meaning is that it cannot be that A is true and B is false. From the connection we saw above, it also follows that B is a sufficient condition for A and A is a necessary condition for B.

Necessary and sufficient condition

There is also a third kind of condition (or another meaning of the thick arrow above): a necessary and sufficient condition. We say that A is necessary and sufficient for B (and equivalently, B is necessary and sufficient for A). This means two things: (1) A cannot happen without B (so B is necessary for A), and (2) if A happens, B necessarily happens as well (so B is sufficient for A). This is symmetry; in fact we have equivalence, and the condition is unique.

It is easy to see that this relation is symmetric. On the one hand, we know that B is necessary for A; from this it follows that B is also sufficient for A. We also know that A is sufficient for B; hence A is also necessary for B. Therefore A is necessary and sufficient for B, and vice versa. The relation is thus denoted A ↔ B, or sometimes A ≡ B.

If you think about it a bit, the meaning of this relation is that A and B always appear together: if one is present (true), then so is the other; if one is absent (false), so is the other. Hence the relation is symmetric with respect to A and B.

The third property of a necessary-and-sufficient condition is uniqueness: if B is a necessary and sufficient condition for A, it is unique. Why? Because if there were another condition C that was also necessary and sufficient for A, then A would not require C in order for the effect to follow (since B suffices for A, it entails A even without C), and if C were absent, A would still occur (because B is present); and if B were absent, then A would not occur even if C were present (since B is necessary). Therefore such duplication is impossible.

The dispute about causality

Philosophers disagree about the nature of the causal relation. All agree that a cause must be a sufficient condition; it is not enough that it be merely necessary. Yet there are two views on how to interpret the causal relation: Is the cause a sufficient condition for the effect (i.e., there is a logical implication A → B), or is it a necessary-and-sufficient condition (A ↔ B)?

At first glance, the second view seems very odd. First, it would make causality symmetric (contrary to our intuition that causality is not symmetric—the cause produces the effect, not vice versa). Second, if the cause is necessary and sufficient for the effect, then, as we saw above, it would follow that there cannot be any other cause for the same effect. But in nature we know many events that can have several distinct causes. For example, the creation of fire can be due to striking a match or focusing the sun’s rays through a magnifying glass; wet ground can be due to rain or to sprinklers. Is it not right to say that striking the match is the cause of the fire, or that rain is the cause of the wetness? This does not seem to comport with our intuitive notion of causality.

One could try to say that striking the match and focusing the sun’s rays are only pre-causes, and that in the end a single thing—heat—is generated, which ignites the fuel and creates fire. That is, at the final stage before the event we define as the effect (the fire), there is indeed only one cause (the heat), though that heat can arise from several sources (match or magnifying glass). The same with wetness: what produces it is the arrival of water to the ground, which likewise can come from different sources (rain or sprinklers).

But even this does not solve the problem, since we saw that there can be several causes even for that final stage (the arrival of water), and therefore it cannot be unique. Thus it is incorrect to define the cause as a necessary-and-sufficient condition.

What we have seen is that while there may exist causes that are both necessary and sufficient for the effect in particular cases, not all causes must be such. This cannot be the general definition of the causal relation. The reasonable conclusion, and the one that better fits our intuition, is that a cause is a sufficient condition for its effect (and in some cases it may also be necessary).

The “kick-start” paradox

Yuval Steinitz, in the second part of his book The Tree of Knowledge, brings this philosophical dispute and proposes a paradox that he believes decides in favor of the first view (that a cause is a sufficient, not necessarily a necessary, condition). He relies on two properties of the necessary-and-sufficient condition we saw above: symmetry and uniqueness.

Suppose we are given that A is the cause of B, i.e., A ⇒ B, and our assumption is that the arrow expresses a necessary-and-sufficient condition. Steinitz argues that if this were the nature of the causal relation, then a causal chain longer than two links could not exist; that is, there could not be A ⇒ B ⇒ C. Proof: if such a chain existed, then A would be the cause of B and B the cause of C, which—since the arrow is necessary and sufficient—by symmetry would let us flip the second arrow and conclude C ⇒ B. But we also know B ⇒ A. Thus both B ⇒ A and C ⇒ B hold, meaning B is necessary and sufficient for both A and C, contradicting uniqueness. Hence no causal chain longer than two links is possible. It would follow further that if A causes B, then B cannot itself cause anything else—which is absurd, since in reality events caused by prior events often serve as causes for subsequent events, typically in an endless causal chain that never actually stops.[1]

Notice that this paradox leads to a stronger conclusion than the one we reached earlier from considering multiple causes. There we saw that a cause need not be unique (i.e., need not be necessary and sufficient). Here, the kick-start paradox seems to rule out the very possibility of a cause that is necessary and sufficient for its effect in any case. It concludes that no cause can be a necessary-and-sufficient condition for its effect. This is a very far-reaching conclusion, and I will now try to show why it is incorrect: a condition can be sufficient but not necessary for the effect.

A critique of the kick-start paradox

First, we must note that in the reversal used above, B is presented as a (necessary-and-sufficient) condition for C. But saying “B is the cause of C” does not mean B precedes C in time, since B arrives before it in the temporal order (we saw that logic is indifferent to time). In the physical plane, given the temporal order, it is more reasonable to say B is the result of A. The causal relation is asymmetric in time, but symmetric logically: the effect is a logical condition for the cause, just as the cause is a logical condition for the effect.

To understand the mistake, assume for the sake of argument that the cause is indeed a necessary-and-sufficient condition for the effect, and assume we have a chain of three events with causal relations: A ⇒ B ⇒ C. Each of these is necessary and sufficient for the next. That means that if A occurs, B must follow, and then C must follow. If so, we were imprecise in saying that A is necessary and sufficient for B and B for C each on its own. What is actually necessary and sufficient for C is the combination of A followed by B. In other words, it may be that B occurs without A, but then C will not occur; and conversely, A may occur without B, and then again C will not occur. In short, if these are necessary-and-sufficient conditions arranged in a chain, there is dependence among them: if C occurs, then necessarily B will occur (and in the past A occurred); there is no possibility of C without B, or B without A. The uniqueness property then characterizes the whole set of events that forms the necessary-and-sufficient condition for C, not each link separately. This again sharpens the difference between a causal relation and purely logical conditioning, if only with respect to the time axis: B is part of the necessary-and-sufficient condition for C, yet it occurs after A and therefore certainly cannot be considered part of the cause of A.

Conclusion

Let me sharpen my earlier distinction. The previous difficulty with the view that the cause is necessary and sufficient remains: it implies there cannot be two different and independent causes for the same thing. The example was striking a match and focusing sunrays, either of which can start a fire without the other. In such a case, clearly neither is a necessary-and-sufficient condition for the effect, since that would contradict uniqueness. If striking a match can ignite a fire without focusing sunrays, then focusing sunrays is not necessary—and conversely. Note that each of these is a sufficient condition (given the circumstances) for the fire, but not a necessary one. From this it follows that a cause is a sufficient, not a necessary, condition for its effect. This matches our earlier conclusion, even though Steinitz’s argument arrived at it by another path.

We will now see an implication of this view (the logical component of the causal relation) for our topic.

Implication: Divine involvement in the world

In Column 297 I discussed Rabbi Moshe Roth’s argument regarding divine involvement in the world. Popularly, divine involvement is expressed in miracles, i.e., in violations of the natural order. But there is also a view that allows for divine involvement within nature, i.e., without a miracle: God “maneuvers” nature to bring about the result He desires. Given that meteorology is a field we cannot predict well, it is tempting to attribute the occurrence or non-occurrence of rain to divine involvement in response to our deeds (mitzvot or sins). As is well-known, the “key of rain” is in God’s own hand and not a messenger’s. Because of the field’s complexity, it is very easy to attribute every event (rain or no rain) to a divine hand.

But as I argued there, if we take seriously the claim that there are no gaps in nature, and that a cause is a sufficient condition for its effect, then given a well-defined set of circumstances, its natural result is determined; no other outcome is possible within the laws of nature. The laws of physics are deterministic in this sense (chaotic processes are deterministic in this sense as well; I have explained more than once why chaos does not depart from a fully deterministic conception), and quantum theory does not help either. In short, quantum theory does indeed speak of such gaps (though that is interpretively disputed), but only at very small scales, irrelevant to daily life. Moreover, according to quantum theory, given certain circumstances there is a defined distribution of possible outcomes; divine involvement would constitute a deviation from that distribution, and therefore still a departure from the laws of nature.[2]

The upshot is that if a given set of climatic circumstances prevails—even if we do not know it fully because it is complicated—it nevertheless dictates the outcome (whether there will be rain). Our deeds cannot dictate rain. Nor does it help to go backward and claim that our deeds ten years ago dictated the circumstances now prevailing and thus the rain now, since those past climatic circumstances were themselves determined by the laws of nature. The climate created by them is indeed complex, but so far as we know it consists of nothing but physics.

Of course, nothing is impossible for God: He can violate the laws of meteorology and climate and bring rain whenever He wishes; the mouth that forbade is the mouth that permitted. But note that if this happens, by definition it is outside nature, a violation of the laws of nature, not “in-nature” involvement. And, as a rule, that is not what happens: forecasters are generally right, and their forecasting window improves with scientific and technological progress. Such involvements—if they exist at all—would be extremely sporadic (this cannot be ruled out, though there is no indication it actually happens).

The essence to understand here is the claim that circumstances together with the laws of nature are a sufficient condition for the outcome. If they are given, it is impossible that the outcome should fail to occur (i.e., that some other outcome would occur). And therefore, if God intervenes to produce a different result than what was expected to occur, He thereby departs from the laws of nature. He can of course do this—but only as a departure from nature, not within it.

See also: https://mikyab.net/en/posts/66508.

This is not the same as what we often mean by “divine involvement.” Usually, we fear that X is about to occur and we prefer Y instead, so we pray that God will bring about Y (i.e., that X will not occur). But if Y was going to occur anyway according to the laws of nature, then there is no need for His involvement. Involvement is called for only where nature does not lead to the desired outcome. For example, a sick person for whose recovery we pray: if he would have recovered without the prayer, then the prayer was unnecessary. The prayer is directed at a case where, by the natural course of things, he was not going to recover, and we ask God to heal him nonetheless. This is a prayer that He intervene and deviate from nature. In most medical cases known to us, the outcome is expected and explicable by medicine; therefore, even if the patient recovered, we do not view it as an open miracle. We cannot determine that there was no hidden miracle, but in essence, if such divine involvement occurred, it was necessarily a departure from the laws of nature.

A closing note: “Everything is in the hands of Heaven”

The view that there can be divine involvement within nature in effect undermines the idea that the cause is a sufficient condition. If the laws of nature and the circumstances are sufficient, then once the circumstances are fixed, the outcome is fixed as well. The consequence is that divine involvement within nature is impossible. Any involvement is necessarily a departure from nature.

By the way, from here it also follows that one cannot truly say that “everything is in the hands of Heaven,” meaning that everything that happens here is in God’s hands. Sporadic involvement may be possible (even if we have no indication that it occurs), but if and when it happens it necessarily departs from the laws of nature. To say that everything is in His hands would mean that there are no laws of nature at all—that everything that happens depends only on the calculus of mitzvot and sins and sundry theological considerations. The laws of nature become a façade, devoid of substance. Such a claim is entirely implausible, as it directly contradicts physics. One can of course say that the phenomenon of gravity is generated by God (with or without the force of gravity), but that does not mean there are no natural laws. It only means the laws of nature are His doing—either from the past (He created them and from then on they operate by themselves) or even in their present existence (everything that happens happens by His current sustenance)—but when we say “everything is in His hands,” we mean “all our happenings are nothing but miracles” (as Nachmanides writes at the end of Parashat Bo), with no nature and no mazal at all. On that view, a person with a fever who takes acetaminophen has his temperature go down because of his morning prayer or his good (or bad) deeds, rather than because of chemistry. A discussion of the naïve doctrine of “hishtadlut” (effort)—why take the acetaminophen at all, then?—would lead me too far afield (see, for example, Column 279). I hope you will allow me not to enter into that here.

See also: https://mikyab.net/en/posts/65479.

For more on the logic involved, see the Q&A here: https://mikyab.net/en/%D7%A9%D7%95%D7%AA/%D7%9C%D7%95%D7%92%D7%99%D7%A7%D7%94-3.

Notes

[1] Perhaps aside from pathological situations in which there is a causal chain of only two links that breaks after the second.

[2] This topic connects to the discussion of parallel planes of reference, which I addressed at length in the second book of the quartet; I will not enter into it here.