On Atomic and Compound Propositions – Between Positive Commandments and Prohibitions (Column 202)

With God's help

In Column 200 we dealt with the logical polygraph by means of which one can extract the truth from any liar. In the course of that discussion I distinguished between compound propositions and atomic propositions: a compound proposition is a complex statement, whereas an atomic proposition is an atomic statement. A compound proposition is made up of atomic propositions that are connected to one another by logical connectives. For example, the proposition "If the sun is currently shining, it is not raining" is made up of two atomic propositions that are connected to one another ("The sun is currently shining" and "It is not raining") by an implication connective. In this column I want to delve a bit more deeply into that distinction, and to see that it has interesting halakhic and meta-halakhic implications.

The difference between a positive commandment and a prohibition: the performance-based definition

The distinction between positive commandments and prohibitions is well known and widely accepted. It even has halakhic implications: to avoid violating a prohibition, a person must give up all his wealth, whereas for a positive commandment only up to a fifth. Violating a prohibition often incurs punishment in a religious court, whereas neglecting a positive commandment never does. A positive commandment overrides a prohibition. Human dignity overrides a positive commandment but not a Torah-level prohibition, and so on.

If you ask people what exactly the difference is between a positive commandment and a prohibition, you will probably receive an answer on the level of performance: a positive commandment imposes on us a duty to act, whereas a prohibition imposes on us a duty to refrain (inaction, omission). For example, the obligation to put on tefillin is a positive commandment because it imposes on us the duty to do something (= put on tefillin). By contrast, the obligation not to eat pork is a prohibition because it imposes on us the duty to refrain from something (= not to eat pork).

More precisely: a positive commandment is fulfilled through positive action and nullified through passive omission. A prohibition, that is, a negative commandment, is fulfilled through passive omission and nullified through positive action. This is the performance-based definition, and that is the accepted way of understanding the distinction between a positive commandment and a prohibition.

The difficulty with the performance-based definition

But the performance-based definition is challenged by quite a few commandments that clearly do not fit it. For example, there is a positive commandment to rest on the Sabbath. How is this positive commandment fulfilled? Through passive omission, of course (by not performing labor). But if it is fulfilled through passive omission and nullified through positive action (when one performs labor), why is it a positive commandment at all? Ostensibly it should be a prohibition. Alternatively, the prohibition of do not stand idly by your neighbor's blood ("do not stand idly by your neighbor's blood") imposes on us a duty to save our fellow when he is in distress. This is a duty of action, meaning that this commandment is fulfilled through positive action and nullified through passive omission. So why is it a prohibition and not a positive commandment?

Paired prohibitions and positive commandments

In the Sixth Root, Maimonides discusses how we are to count commandments in which there is a paired prohibition and positive commandment (imposing the same obligation). For example, the prohibition and positive commandment regarding labor on the Sabbath and festivals, returning lost property, fasting on Yom Kippur, sending away the mother bird, building a parapet, giving charity, and the like. It should be noted that in every case of such a pairing there must necessarily be a deviation from the performance-based definition. If the obligation is to do something, then the prohibition here will be the exceptional case (it will be fulfilled through positive action and nullified through passive omission), and if the obligation is to refrain, then the positive commandment will be the exceptional case (for it will be fulfilled through passive omission and nullified through positive action).

Maimonides defines the principle there as follows:

The sixth root: a commandment that contains both a positive commandment and a prohibition, it is proper to count its positive element among the positive commandments and its prohibition among the prohibitions.

In our essay on the Sixth Root,[1] we pointed out that these words contain two innovations:

• That in a pairing of a prohibition and a positive commandment, both commandments are counted (as opposed to the Ninth Root, where Maimonides rules that in a duplication of two prohibitions or two positive commandments, only one commandment is counted).
• That the prohibition is counted among the prohibitions and the positive commandment among the positive commandments.

At first glance, it is not clear what the novelty is in point b, since it is obvious that we are dealing here with a prohibition and a positive commandment. But in light of what we said above, it is clear why Maimonides had to spell this out. We saw that in every case of pairing there is necessarily an exceptional commandment. Therefore, one might have said that even if we count a prohibition and a positive commandment regarding refraining from labor on the Sabbath, they should be counted as two prohibitions, since both impose on us a duty to refrain and not a duty to act. Alternatively, the prohibition and positive commandment of sending away the mother bird impose on us a duty of action, and therefore the prohibition too should be counted as a positive commandment rather than as a prohibition. Maimonides therefore innovates that beyond the fact that both commandments enter the tally of the commandments (innovation a), we count the prohibition among the prohibitions and the positive commandment among the positive commandments (innovation b). This is an additional innovation, grounded in the conception that the performance-based definition of prohibition and positive commandment is incorrect. There are prohibitions that impose a duty of action, and positive commandments that impose a duty to refrain.

And the difficulty returns

This of course raises the following difficulty: what, then, is the difference between a prohibition and a positive commandment? If each of them can impose both a duty of action and a duty to refrain, why is one a prohibition and the other a positive commandment? What is the principled difference between these two kinds of command? In other words, what exactly is the difference between the positive commandment to rest on the Sabbath and the prohibition against performing labor on the Sabbath? Why is this one a prohibition and that one a positive commandment?

The difficulty with the performance-based definition is empirical: in practice there are commandments that do not fit it. By contrast, the present difficulty is essential: if not the performance distinction, what is the difference between a prohibition and a positive commandment at all?

A linguistic distinction

This question was addressed by the late Prof. Aharon Shemesh, a Talmud scholar from Bar-Ilan University, in his article "On the History of the Meaning of the Concepts Positive Commandments and Negative Commandments" (hereafter: Shemesh).[2] Shemesh argued that the distinction between positive commandments and prohibitions underwent changes during the Talmudic period. In earlier passages the difference was performance-based (and then the commandment to rest on the Sabbath was indeed only a prohibition and there was no positive commandment there), but in later generations the definition became linguistic (and then the paired commandments were added as well), that is, according to the way the Torah formulates the command: if it is formulated as a positive command (positive wording), it is a positive commandment; and if the formulation is negative ("beware," "lest," and "do not"; see Makkot 13b and parallels), it is a prohibition.

The difficulty with the linguistic definition

But this definition too does not really answer our difficulty. After all, the Torah did not choose its wording arbitrarily either. If there is no real difference between a prohibition and a positive commandment apart from wording, why was precisely this wording chosen? Why does the Torah formulate the prohibition against eating pork in negative language rather than positive language? Why does it formulate the obligation to rest on the Sabbath in positive language rather than negative language? Is this an arbitrary choice? Some essential criterion ought to stand behind it. And if that criterion is not the performance-based one (as we have seen), what is it? The linguistic proposal merely pushes the problem back one step. We now ask what underlies the linguistic distinction between a prohibition and a positive commandment. The question of the essential and substantive difference between them still stands.

In my article on Torah and science (here, and briefly here), I pointed out that this is a good example of the difference between the academic approach and the traditional one. The academic approach cleaves to solid facts and avoids speculation, and therefore Shemesh grounds the distinction between a prohibition and a positive commandment in a fact: the Torah's wording. But the traditional learner asks himself: why does the Torah formulate it this way? Surely there must be some essential difference behind these forms of expression. An answer to such a question necessarily contains an element of speculation, because we cannot know whether the reasoning we propose is correct or not. All we have is the definition the Torah gives us, and the explanations we find are our own. Therefore an academic scholar is cautious, and rightly so, about engaging such questions. On the other hand, these are important questions, and without them the solid facts remain disconnected.

The question can be broadened to include all the commandments, even those whose definition seems reasonable to us. For example: why is the prohibition against eating pork a prohibition? It could have been written as a positive commandment not to eat pork. And the commandment to put on tefillin is an obligation not to be in a state without tefillin. According to the linguistic criterion, if the Torah writes "Do not be without tefillin" that would be a prohibition, whereas when it writes "Put on tefillin" it is a positive commandment. But these are two commands with the same content, so why should the wording change anything? This is a double negation, and that returns us to the sentence's original meaning. It cannot be that two commands that are logically equivalent should be regarded as two different commandments with different halakhic status (see above the halakhic implications of a given commandment being a prohibition or a positive commandment).

Back to compound and atomic propositions

We now return to the distinction between compound and atomic propositions mentioned at the beginning of the column. We saw in Column 200 that an atomic proposition is an atomic statement from which one builds a compound proposition by means of logical connectives. Thus, for example, the connective ∨ (=or) can combine (connect) two atomic propositions, A and B, and thereby create from them one compound proposition: A ∨ B. If A is the atomic proposition "It is raining" and B is the atomic proposition "There are clouds," then the compound proposition A ∨ B is the sentence "It is raining or there are clouds" (or, in simpler English: "Either it is raining or there are clouds"). Another connective, such as ∧ (=and), will combine those same two atomic propositions into the following sentence: B ∧ A, whose meaning is: "It is raining and there are clouds." Another connective is implication, denoted →, and the meaning of the sentence it forms, A → B, is: if A, then B.

All these connectives are binary connectives, that is, they connect two atomic propositions and create from them a compound proposition. But there is one exceptional connective, which is unary: the negation connective, denoted ¬. The meaning of the sentence formed by it, ¬A, is: not A (that is, it is not true that A). It is important to understand that such a sentence is not an atomic proposition but a compound proposition. A is an atomic proposition, but ¬A is a compound proposition (made up of the atomic proposition A and the negation connective).

Defining commandments by means of compound and atomic propositions

A positive commandment instructs us to do something, for example to put on tefillin. Let the act of putting on tefillin be X. The command to put on tefillin is something like: "I command you to do X." The command, of course, turns this sentence into something that is not a proposition, since a command is neither true nor false. It is therefore better to focus on the content of the command, namely the atomic proposition A: "the agent does X," which is of course a proposition in every respect (it can be true or false). The command instructs the agent to ensure that the proposition A is true.

Now think about the status of a commandment such as: "I command you to do non-X" (= a command not to put on tefillin). Here the relevant atomic proposition is B: "the agent does non-X." Now it is easy to see that this too is a positive commandment, because it can be described by a similar atomic proposition, except that here the action regarding which we were commanded is not X but non-X. But as we saw above, a positive commandment can also command non-performance. If so, we have here a positive commandment that instructs me to ensure that the atomic proposition B is true.

And what about a prohibitory command such as: "I command you not to do X"? At first glance this is exactly equivalent to the previous command: "I command you to do non-X," and therefore this too should be a positive commandment. That was in fact our difficulty above, when we wondered what the difference is between a positive commandment and a prohibition. But now we can see that this is incorrect. To do that, we must again strip away the command and examine only its propositional content. This time we are dealing with the proposition C: "the agent does not do X," or alternatively: "it is not true that the agent does X." Now you can see that this is not an atomic proposition but a compound proposition built from a negation connective together with the atomic proposition A. The formalization is: C = ¬A. Therefore this is not a positive commandment but a prohibition. A prohibition is a command that a compound proposition obtain (that it be true), whereas a positive commandment is a command that an atomic proposition obtain (that it be true).

We have now arrived at a sharp logical definition of prohibition as opposed to positive commandment: a positive commandment is a command to ensure that some atomic proposition (positive or negative) is true. A prohibition is a command to ensure that some compound proposition (positive or negative) is true. When I command you to put on tefillin, or to be in a state in which you do not have tefillin on, that is a positive commandment, because I require of you that a state described by an atomic proposition obtain. By contrast, when I command you not to be in a state in which you have tefillin on, or not to be in a state in which you do not have tefillin on, that is a prohibition, because I require of you that a state described by a compound proposition obtain (and not by an atomic proposition).

Double negation

We can now understand why the problem we raised above regarding the equivalence between a prohibition and a positive commandment is incorrect. The assumption there was that one can formulate a prohibition as the double negation of a positive commandment. For example, if I command, "Be in a state with tefillin" or "Do not be in a state without tefillin," these are equivalent propositions. We therefore wondered how it could be that two propositions that are logically equivalent should have different halakhic status, one a positive commandment and the other a prohibition.

But from what we have seen here it follows that there is no way to formulate a prohibition as the double negation of a positive commandment. Suppose the positive commandment is a command to put on tefillin. The command to be in a state without tefillin is another positive commandment. The next negation is a prohibition against being in a state without tefillin, and that is indeed a prohibition. Is it not equivalent to the original positive commandment? In the original positive commandment we are obligated to put on tefillin, whereas in the negative formulation we must ensure that we are not without tefillin. In practice, that will usually require us to put on tefillin for that purpose, but this is different logical content, and as we shall immediately see, these two commandments also have different normative content.

Interim summary

In fact, we now have several formulations that it is important to distinguish from one another: "I command you to do X," or "I command you to do non-X," are positive commandments (thus, for example, "I command you to do no labor" is the positive commandment to rest). The formulation "I command you not to do X" is a prohibition (for example, the prohibition against doing labor). The following two sentences, "I do not command you to do X" or "It is not true that I command you to do X," are not imperative sentences at all but declarative sentences. They describe a state of affairs (the claim is that there is no such command) rather than commanding or warning.[3]

Let us now consider the formulation of the prohibition: "I command you not to do X," as opposed to the positive commandment that commands refraining: "I command you to do non-X." These two look very similar. In practice, the positive commandment to rest on the Sabbath and the prohibition against performing labor are seemingly two commands that instruct us to do the same thing. But that is not precise. For example, if I was coerced and performed labor on the Sabbath (for example, in a life-threatening situation), then I did not violate the prohibition, but clearly I did not fulfill the positive commandment. If I purchased a house that already had a parapet, then I did not violate the prohibition of do not place bloodguilt in your house, but I did not fulfill the positive commandment of make a parapet for your roof. The root of the difference is that the prohibition requires only that I not violate it (one can violate it, but not fulfill it), whereas the positive commandment requires fulfillment (it can be fulfilled, and usually it can also be neglected—failure to fulfill a positive commandment—unless it is an existential positive commandment).

Explanation: love and fear

The root of the difference between a prohibition and a positive commandment is explained in our essay on the Sixth Root (where various sources for this distinction are also cited). In brief, we explained there that a positive commandment directs us toward a positive state, whereas a prohibition warns us against a negative state. If there is a positive commandment to rest on the Sabbath, that means that rest is a positive state. Performing labor is not a negative state; it is simply absence of the positive state. But when there is a prohibition against labor on the Sabbath, that means that performing labor is negative, whereas rest is not positive but only absence of the negative state. Therefore it makes no difference whether the positive commandment instructs us to act or to refrain. In the case of a positive commandment, the action or the refraining are positive states in which we are meant to be. When the action is a negative state and we are therefore required to refrain from it, that will be a prohibition. And the same applies if the negative state against which we are warned is a state of refraining (and to avoid it one must do something, as in do not stand idly by your neighbor's blood).

Nachmanides, in his remarks on the commandment of rest on the Sabbath in Parashat Yitro, defines prohibitions as stemming from the quality of fear, and positive commandments as stemming from the quality of love. The prohibition warns against doing something that is bad in the eyes of the One who commands, and obedience to it is rooted in fear of the Commander, whereas the positive commandment is the doing of what is good in His eyes, which is based on love of the Commander.

Back to the distinction between positive action and passive omission

We explained there that this distinction implicitly assumes the existence of some "zero line" that separates positive from negative states. The definition of the states does not derive from the Torah's command; rather, they have an objective definition. Hence a command to be in a positive state does not mean merely not to be in a negative state. The implication is that when we find ourselves in the zero state itself (as, for example, when labor is performed under duress), then we have not fulfilled the positive commandment, but neither have we violated the prohibition.

From here all the differences between a prohibition and a positive commandment can be derived, and yet we can still see that none of this has any necessary connection to positive action or passive omission. The punishment for a prohibition is not because one transgresses it through positive action, but because violating it means being in a negative state. By contrast, not being in a positive state does not incur punishment, but only withholding of reward. The same applies to the obligation to give up all one's wealth in order to avoid a prohibition, and so forth.

But if we look again at the picture described here, we can see that the association people make with the level of performance is not unfounded. When a person violates a prohibition, he is in a negative state, that is, he collides head-on with the will of the Torah. When a person neglects a positive commandment, he is simply not in a positive state; that is, there is no head-on collision with the will of the Torah, but only failure to fulfill it. For example, when the Torah commands us to rest on the Sabbath, one who performs labor does not collide head-on with its will, but merely fails to fulfill it. But when the Torah forbids labor through a prohibition, one who performs labor acts head-on against its will. The conclusion is that violation of a prohibition, even if done by omission, is an act that stands in direct opposition to the will of the Torah, and therefore in a certain sense it is a transgression of positive action. By contrast, neglect of a positive commandment, whether done by deed or by omission, is an act that does not collide with the will of the Torah, and therefore in an essential sense it is a transgression of passive omission.

The terms positive action and passive omission undergo abstraction here. A transgression of positive action is not necessarily active performance on the level of behavior, but the clash with the will of the Torah is active. I am "doing" in the sense of acting against the will of the Torah. A transgression of passive omission is not necessarily an omission in the behavioral sense, but it is an omission in the essential sense. If we return to Shemesh's description, we actually have here another example of a process of abstraction that is very characteristic of Jewish law in the transition from the earlier generations to the later ones. Scholars have already pointed out that the Talmud tends to take the Mishnah's concrete definitions and abstract them. That process, of course, continues among the medieval authorities and later authorities. As we saw here, the distinction between a prohibition and a positive commandment is a good example of such a process.

[1] An updated version of it appears in the book Yishlach Sharashav, which is also available on the website, in the essay on the Sixth Root. See there for a long and detailed discussion of the distinction between a prohibition and a positive commandment and its implications, of which only the main points appear here.

[2] Tarbiz 72, issues 1-2, 2003.

[3] See our essay on the Eighth Root, where we discussed the negation of obligation and explained why, according to Maimonides, this is not a command at all.