Causality: C. A Look at the Logical Component of Causality (Column 462)
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/translated-articles-rabbi-michael-abraham/post-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/translated-articles-rabbi-michael-abraham/post-65479.
For more on the logic involved, see the Q&A here: https://mikyab.net/%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.
Contents of the Article
With God’s help
Causality: 3. A Look at the Logical Component of Causality
In the previous two columns I argued that the causal relation has three components: the temporal, the logical, and the physical, and I noted the difficulty of treating the physical component formally (that is, symbolizing it). Now I would like to enter a more precise analysis of the logical component, and to some extent also of the temporal component. I will begin the discussion with a logical distinction between different kinds of conditions.
A necessary condition and a sufficient condition
First, let us denote the logical condition in the form of an implication (though not necessarily in the material sense; this is only notation for our purposes): A 🡺 B. Note that this arrow parallels the double arrow from Column 459. A is the antecedent of the conditional (that which implies), and B is its consequent (that which is implied). Logicians distinguish between two kinds of conditions: a necessary condition and a sufficient condition. These are two different meanings of the arrow 🡺 that appears in the implication relation above.
A necessary condition. When we say that A is a necessary condition for B, we mean that B cannot occur unless A obtains. That is, the existence of A is necessary for B.
Notice that the obtaining of A may be earlier than B in time or later than it, for as we saw in the first column, logic is indifferent to the time axis (this is another reason not to identify it with causality). Let us see this by means of an example. One may say that clouds are a necessary condition for rain, since rain cannot fall unless there are clouds. By the same token, one may say that the wetness of the ground is a necessary condition for rain, since it cannot be that rain is falling and the ground is not wet. In the second case it is clear that we are not dealing with a cause, because the wetness of the ground comes later than the rainfall. In causal terminology we would say that it is the result of the rain, not its cause. A cause must precede its effect, but in a relation of conditionality the condition does not necessarily precede that which is conditioned.
What characterizes both of these examples is that the antecedent does not necessarily imply the consequent; that is, the existence of clouds does not necessarily imply rain (if these are not rain clouds, or if the circumstances prevent rain from forming), just as the wetness of the ground does not necessarily imply rain (perhaps there were sprinklers).
We shall now define another meaning of the implication arrow: a sufficient condition.
A sufficient condition. When we say that A is a sufficient condition for B, the meaning is that if A obtains, B necessarily obtains. In such a case, it is enough to know that A in order to infer B.
If we examine the two examples I gave above, we will easily see that if A is a necessary condition for B, then B is a sufficient condition for A. For example, the knowledge that it is raining is sufficient to infer the conclusion that there were clouds earlier. By the same token, the knowledge that it rained is enough to infer that the ground was wet afterward. Thus, rain is a sufficient condition for both of these outcomes.
More generally, this can be proven by reductio. Let us assume that A is a necessary condition for B. From here one can prove that B is a sufficient condition for A. Assume the opposite, namely, that B obtains and yet A does not. But A is a necessary condition for B, so if B obtains, A must obtain as well. Therefore it cannot be that B obtains and A does not. Q.E.D. The converse proof is of course identical: if A is a sufficient condition for B, then B is necessary for A.
A note on material implication
The logical implication relation, A🡪B (this is not the causal arrow that I marked with a double or thick arrow, but a logical implication), can be defined in several ways. The accepted way to define it is materially (see the explanation here), that is, that it is impossible for A to be true and B false. From the definition above it follows that the meaning of material implication is that A is a sufficient condition for B. From the connection we saw above, one may also infer from this that B is a necessary condition for A.
A 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. When we say that A is a necessary and sufficient condition for B, this means that both relations obtain between them (A is both necessary and sufficient for B): not only is it impossible for B to occur without A (A is necessary for B), but also, if A obtains, then B necessarily obtains as well (A is sufficient for B).
We shall now see that the relation between A and B, when A is a necessary and sufficient condition for B, has three important properties: the relation between A and B is symmetric; in fact, it is an equivalence; and the condition A is unique.
It is easy to see that this relation is symmetric. On the one hand, we know that A is necessary for B. Above we saw that this means that B is a sufficient condition for A. We also know that A is sufficient for B, and from this it follows that B is necessary for A. Thus, when A is necessary and sufficient for B, B too is necessarily necessary and sufficient for A; that is, this is a symmetric relation. Therefore a necessary and sufficient condition between A and B is denoted by a bidirectional arrow: A ↔ B.
If you think about it a bit, you will see that the meaning of this relation is that A and B always appear together; that is, if one of them obtains (is true), then the other obtains as well, and if one of them does not obtain (is false), then the other does not obtain either. Notice that I did not specify which is one and which is the other, and that is because this statement is symmetric with respect to A and B. Therefore, a relation of necessary and sufficient condition is sometimes also denoted as equivalence: A ≡ B.
The third property of a necessary and sufficient condition is that it is unique. By this I mean that if A is a necessary and sufficient condition for B, there cannot be another condition C that is also necessary and sufficient for B. Why? Because if A obtains, then C is not needed in order for the effect to emerge (since A is a sufficient condition, and therefore it implies B even if C does not obtain), and if A does not obtain, then it will not help that C obtains, and B necessarily does not obtain (since A is necessary for B).
The debate over causality
Philosophers disagree about the nature of the causal relation. Everyone agrees that a cause must be a sufficient condition, and it is not enough for it to be merely a necessary condition. Still, there are two views regarding the interpretation of the causal relation: is the cause A a sufficient condition for the effect B (that is, does a logical implication hold between them: A🡪B), or is it a necessary and sufficient condition for it (A ↔ B)?
At first glance, the second conception seems very strange. First, it appears to imply that the causal relation is symmetric, although intuitively it is clear to us that causality is not symmetric (the cause produces the effect, not the other way around). Second, if the cause A is a necessary and sufficient condition for the effect B, then from what we saw above it follows that there cannot be another cause, C, of the same effect. But we know of many events in nature that may have several different causes. For example, the formation of fire may result from striking a match or from concentrating the rays of the sun by means of a magnifying glass. The wetness of the ground may result from rain or from sprinklers. Is it not correct to say that the rain is the cause of the wetness, or that striking the match is the cause of the fire? This does not seem to fit our intuitive understanding of the concept of causality.
Seemingly, one might say that striking the match and concentrating the sun’s rays are only earlier causes, but in the end heat is produced there, which ignites the combustible material and creates fire. That is, at the final stage before the event defined as the effect (the fire), there is indeed only one cause (the heat), but that cause (the heat) can perhaps be produced by several different sources (a match or the focusing of the sun’s rays). The same is true of wetness: what produces it is always the arrival of water at the ground. But the arrival of the water can be produced by several different sources (such as rain or sprinklers).
But even this does not solve the problem, for one can regard this last cause (the heat, or the arrival of water at the ground) as the effect, and now ask what its own cause is. We saw that this effect may have several different causes. Thus, once again we arrive at the conclusion that a cause need not be unique, and therefore it is incorrect to define a cause as a necessary and sufficient condition for the effect. Let me sharpen the point: there may be causes that are necessary and sufficient, but what we have seen here is that not all causes must be such. Therefore, this cannot be the definition of the causal relation.
The conclusion is that the reasonable definition, and the one more suited to our intuition, is that the cause is a sufficient condition for the effect (and in certain cases it may also be necessary).
The initiation paradox
Yuval Steinitz, in the second part of his book The Tree of Knowledge, presents this philosophical dispute and proposes a paradox that would decide in favor of the first conception (that a cause is a sufficient condition that need not be necessary). He relies on two properties of the necessary and sufficient condition that we saw above: symmetry and uniqueness.
We are given that A is the cause of B, that is, A 🡺 B, and our assumption is that the arrow expresses a necessary and sufficient condition. Steinitz argues that one can prove that if this were the essence of the causal relation, then a causal chain of more than two links could not exist; that is, the following could not obtain: A 🡺 B 🡺 C.
Proof: if there were such a chain, then A is the cause of B and B is the cause of C. In other words, the following two relations would obtain here: A 🡺 B and B 🡺 C. But if the arrow is a necessary and sufficient condition, then because of the symmetry of this relation one may reverse the direction of the second arrow, that is, C 🡺 B would also obtain. But we also know that A 🡺 B obtains. This means that we have two necessary and sufficient conditions for B (A and C), which contradicts the property of uniqueness. We have reached a contradiction, and this means that a causal chain of more than two stages is impossible. If A is the cause of B, then B cannot be the cause of anything else. But that itself, of course, cannot be, for there is nothing to prevent an event caused by another event from being the cause of a third event. On the contrary, the events we know are usually part of an endless causal chain that never ceases.
Notice that this paradox leads to a more far-reaching conclusion than the one we reached in the previous sections. Above we saw, by reflecting on examples, that a cause need not be unique, that is, it need not be a necessary and sufficient condition. But we left open the question whether there can in principle be a case of a cause that is a necessary and sufficient condition for its effect. There we saw that, in principle, this should not be ruled out. The initiation paradox rules this out, since it leads to the conclusion that there can be no case at all of a cause that is a necessary and sufficient condition for its effect. A cause, as such, is only a sufficient condition, but not a necessary one, for the effect. This is a very far-reaching conclusion, and I will now try to show why it is incorrect.
A critique of the initiation paradox
First, we must note that in the reversal we made above, C is presented as a condition (necessary and sufficient) for B. This likewise does not mean that C is the cause of B, since it comes later than B in time (we saw that logic is indifferent to the time axis). What, then, is the relation between it and B on the physical plane? In light of the temporal order, it is more reasonable to say that it is the result of B. The result is also a logical condition for the cause, just as the cause is a logical condition for the result. The causal relation is temporally asymmetric, but it is symmetric on the logical plane.
To understand Steinitz’s mistake, let us assume for the sake of the discussion that the cause is indeed a necessary and sufficient condition for the effect. Let us also assume that there is a chain of three events with cause-and-effect relations: A 🡺 B 🡺 C. Each of these is a necessary and sufficient condition for the one that follows it. This means that if A occurred, then B will necessarily occur after it, and after that C will necessarily occur. All of these are entirely necessary. If so, we were not precise when we said that A is a necessary and sufficient condition for B. What is actually necessary and sufficient for B is the combination of A and C together. In other words, I am claiming that B cannot occur unless A is before it and C is after it. I refer you to the proof of the uniqueness property of the necessary and sufficient condition. There I said that if A obtains, then B must occur even without C obtaining (since A is sufficient), and if A does not obtain, then even if C does obtain, B will not happen. But that argument assumes independence between A and C. Yet if we are dealing with a chain of necessary and sufficient conditions, then there is certainly dependence. If A occurs, then in the future C will necessarily occur as well. There is no possibility that A obtain without C, or C without A. In other words, even if we accept the property of uniqueness, it characterizes the totality of the events that constitute the condition for B, and not each of them separately. This again sharpens the difference between a causal relation and logical conditionality, if only from the standpoint of the time axis. C is part of the necessary and sufficient condition for B, even though C occurs after it and therefore certainly cannot be regarded as part of its cause.
Conclusion
Let me sharpen further the distinction I made above. It is easy to see that the earlier difficulty I raised concerning the causal conception according to which the cause is a necessary and sufficient condition still stands. We saw that it implies that there cannot be two different and independent causes of the same thing. The example of this was striking a match and focusing the sun’s rays, each of which without the other (!) can cause fire. This is an example of two events independent of one another (each of which can appear without the other), and in such a situation it is clearly impossible to view either of them as a necessary and sufficient condition for the effect, because that contradicts the property of uniqueness of such a condition. If striking a match without focusing the sun’s rays can ignite a fire, that means that focusing the sun’s rays is not necessary, and vice versa. Notice that each of these is indeed a sufficient condition (given the circumstances) for the creation of the fire, but not a necessary one.
By virtue of this argument, the conception that the cause is a sufficient and not a necessary condition for the effect is indeed compelled.[1] Notice that although Steinitz’s initiation paradox, which tried to prove this in another way, is an invalid argument, its conclusion is correct. For our purposes, the logical component of the causal relation is that of a sufficient condition. We shall now see an implication of this conception.
Implication: divine involvement in the world
In Column 297 I discussed Rabbi Moshe Rat’s argument regarding divine involvement in the world. Divine involvement in the world is expressed in a miracle, that is, in a deviation from the laws of nature. But there is a widespread conception according to which divine involvement may also occur within nature, that is, without a miracle. The Holy One, blessed be He, manipulates nature in such a way as to lead to the result He desires. Thus, for example, meteorology is a field that we do not know how to predict very well, and therefore ostensibly there is no obstacle to claiming that rainfall is a product of divine involvement following our actions (commandments or transgressions). As is well known, the key to rain is entrusted to the Holy One, blessed be He Himself, and not to any messenger. Because of the complexity of this field, it is very easy to attribute any event (there is rain or there is no rain) to the divine hand.[2]
In my discussion there I explained that this is impossible, if one takes into account the assumption that there are no gaps in nature, and that a cause is a sufficient condition for the effect. Given a well-defined system of circumstances, its natural result is necessitated; that is, no other result is possible within the framework of the laws of nature. All the laws of physics are deterministic in this sense. I have explained more than once why chaos is not a deviation from this conception (a chaotic process is completely deterministic), but neither is quantum theory. In brief, quantum theory does indeed speak about such gaps (and even this is a matter of interpretive dispute), but this happens on tiny scales, ones that are irrelevant to our everyday lives. Moreover, according to quantum theory, under given circumstances there is a defined distribution of the different possibilities (the probability that each of them will occur), and divine involvement constitutes a deviation from that distribution, so this too is still a deviation from the laws of nature.
The meaning of this is that if a given system of climatic circumstances prevails, then even if we do not fully know it because it is complex, it dictates the result (whether there will or will not be rain). This means that within the framework of the laws of nature, our actions cannot dictate the rainfall. There is also no point in going backward, that is, in assuming that our actions ten years ago dictated the circumstances that now prevail and hence also the rain that arises from them, for those climatic circumstances too were determined deterministically by the laws of nature. The climate is indeed complex, but as far as we know it contains nothing beyond physics.
Of course, nothing is beyond the Holy One, blessed be He, and He can also violate the laws of meteorology and climate, and make rain fall whenever He wishes. He created the laws of nature, and He can suspend them as well. The same mouth that prohibited is the mouth that permits. My claim here is that even if this happens, it necessarily involves a deviation from the laws of nature, and not involvement within them. Moreover, even if this happens, it is plainly evident that this is not what usually happens. There is no visible connection between prayers and rain, or between good or bad deeds and rain. Usually the forecasters are right, and their forecasting range has been steadily improving over the years with the advance of scientific and technological knowledge. Such involvements, even if they exist, are at most sporadic cases (and this cannot be ruled out, although there is no indication that it happens).
One must understand that this is in fact the meaning of the claim that the circumstances and the laws of nature constitute a sufficient condition for the result. If they are given, it cannot be that the result will not occur (that is, that a different result will occur). And from this it follows that if the Holy One, blessed be He, intervenes and causes a result different from what was expected to occur, then by that very fact He has deviated from the laws of nature. Again, He can of course do this, but only by way of a deviation from nature. There is no divine involvement within nature.
Think for a moment, on the basis of the above, about what we call divine involvement. We fear that X is supposed to happen and that this is inconvenient for us, and so we ask the Holy One, blessed be He, to cause Y to happen instead (that is, for X not to happen). If, in any event, Y was supposed to happen according to the laws of nature, then there is no need for His involvement. Involvement is called for only where nature itself does not lead to the desired result. For example, a sick person for whose recovery we pray. If he would recover even without the prayer, then the prayer is superfluous. The prayer is directed toward a situation in which he is not supposed to recover by way of nature, and we ask the Holy One, blessed be He, nevertheless to intervene and heal him. This is a prayer for a deviation from nature. In most cases this is a hidden miracle, for the medical result is not fully known to us. There are cases of unexpected recovery or unexpected illness that medicine does not know how to explain. Therefore, even if the patient recovered, we do not see this as an open miracle, nor can we determine that there was no miracle here. But in essence, if this happened through divine involvement, then it necessarily involved a deviation from the laws of nature.
A final note: “Everything is in the hands of Heaven”
The conception that there can be divine involvement within nature in fact undermines the view that the cause is a sufficient condition. If the laws of nature and the circumstances constitute a sufficient condition, then the moment the circumstances are fixed, the result is fixed as well. The implication is that divine involvement within nature cannot exist. Any such involvement is necessarily a deviation from nature.
Incidentally, it also follows from this that there is no way to say that everything is in the hands of Heaven, that is, that everything that happens here is in the hands of Heaven. Sporadic involvement is possible (even if we have no indication that it occurs), but if it does occur then it necessarily involves a deviation from the laws of nature. But the statement that everything is in His hands means that there are no laws of nature at all. Everything that happens depends only on considerations of commandments and transgressions and various theological considerations. The laws of nature become a facade devoid of real substance. Such a claim is already wholly unreasonable, since it flatly denies physics. One may of course claim that the phenomenon of gravitation is brought about by the Holy One, blessed be He (whether through the force of gravity or even without it), but that does not mean that there are no laws of nature. It only means that the laws of nature are the handiwork of the Holy One, blessed be He, either in the past (He created them, and from then on they operate on their own) or even in the present (everything that happens occurs through the power of the Holy One, blessed be He, and through His present involvement). But when we say that everything is in His hands, we mean that “all our occurrences are nothing but miracles” (in Ramban’s formulation at the end of Parashat Bo), and that there is no nature or fate at all. According to this, if a person has a fever and takes acetaminophen, the reduction of the fever is a result of his prayer in the morning or of his good or bad deeds, and not of chemistry. Well, the question of why he should take the acetaminophen at all leads me to the discussion of the infantile thesis of “human effort” (see, for example, Column 279). I hope you will allow me not to go into that again.
1.
Footnotes
- Perhaps with the exception of pathological situations in which there is a causal chain only two links long, which is cut off after the second link.
- This subject is connected to the discussion of parallel frames of reference, which I treated at length in the second book of the quartet, but I will not enter into it here.
Discussion
I didn’t understand the question. In any case, I hold that there is free choice, and reconciling this with neuroscience is the subject of my book The Science of Freedom. A summary appears in an article here on the site about free will.
A. What you wrote regarding divine intervention is exactly what the Ramban wrote in Torat Hashem Temimah (which expands on what he wrote in Parashat Bo); see there and in several other places where he elaborated on this.
B. When Hazal say, “Everything is in the hands of Heaven,” this can be interpreted as meaning through the laws of nature or mazal, as opposed to in the hands of man. See Tosafot on “Everything is in the hands of Heaven except…”
Only the laws of physics determine causality. Everything else is magical attempts to force things upon reality through words and logical persuasions aimed at the gullible. In practice, it is all imagined in the imagination. And the magic has lost its sting.
Sorry, I didn’t understand this sentence. Maybe it can be explained again?
“One can say that clouds are a necessary condition for rain, since rain cannot fall without clouds. Note that the rain appears after the clouds; that is, their being a condition for rain does not mean that they precede it in time.”
(At the beginning, under the heading: Necessary Condition)
Sorry, there is a mistake here in the order of things. What I meant to say was that rain can be a condition for clouds (a sufficient condition), even though it appears after them in time. To be the cause of something necessarily means appearing before it. To be a condition for something does not necessarily mean appearing before it.
I have now corrected the paragraph:
Note that the occurrence of A can be at a time prior to B or after 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 see this through an example. One can say that clouds are a necessary condition for rain, since rain cannot fall without clouds. By the same token, one can say that wet ground is a necessary condition for rain, since it cannot be that rain is falling and the ground is not wet. In the second case it is clear that this is not a cause, because the wetness of the ground comes later in time than the rainfall (in causal terminology we would say that it is the result of the rain and not its cause). A cause must precede its effect, but with a condition, the condition does not necessarily precede what is conditioned.
Thank you very much. Another question:
“Given a well-defined set of circumstances, its natural result is entailed; that is, no other result is possible within the framework of the laws of nature. The laws of physics are all deterministic in this sense.”
I didn’t understand what “given a well-defined set of circumstances” means—who defined what is given?
(Second paragraph under the heading “Implication: God’s Involvement in the World”)
I mean that the laws of nature operate on a given physical state. If there is an object suspended in the air, the law of gravity determines that it will fall. But if it is already on the ground, then it will not fall. Therefore, to describe the course of events in the world, two things are required: a full description of the given state and the laws of nature. Of course, every such given state is the result of previous states (and human choices).
Where do we have a full description of the given state? After all, we will never know whether what we defined as a sufficient condition, according to our interpretation of the given state, really is in itself sufficient.
This is a fundamental problem in physics and in science generally. But fortunately, in most cases there is a very partial description that is sufficient. For example, when an object is standing in the air, that is a sufficient description to determine that it will fall, even without taking into account all the rest of the data of the universe. Beyond that, even if we do not have a full description, the determinism of the laws still means that the given state (which is not fully known to us) determines the result.
I didn’t understand how you conclude that if there is divine involvement, it violates the sufficient condition. By the same token, you could assume that it is included within it. After all, in any case you do not have the full description.
That has nothing at all to do with whether I do or do not have the full description in hand, but only with the question of whether the laws are correct.
It follows from the nature of the laws of physics. One can of course deny their correctness, but if these are the laws, then they are a sufficient condition and the Holy One, blessed be He, is not part of that.
According to this, is every human choice just a sporadic accident? After all, there is no crossroads at which the neurons stop and act according to a person’s choices; rather, every electrical process in the brain leads to the next one until the final choice, which was caused by all the natural results such as the environment, etc., all as part of the laws of physics. Or perhaps there is no choice at all?