A Look at the Law of the Excluded Middle (Column 660)

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.

Dedicated to my friend Menachem Finkelstein, who sent me these two delightful passages by Agnon

People tend to say: we are masters of our deeds; what we do, we do knowingly. I cannot say so. Everything I do seems to me as though, were it up to me, I would not have done it.

I return to the matter and will tell what happened after a door opened before me and I entered the house—namely, a room—for like most simple houses in our town there was no corridor; a person opens the door facing the street and finds himself standing inside one of the rooms of the house. The room was not small and not large, nor even medium: one of those rooms whose very form makes it hard to gauge their size.

(S. Y. Agnon, “One Day – A Story of Walks,” in Tachrich Shel Sippurim)

There is a certain compartment in Gehenna called Shadow of Death. Longer than the land is wide, and its width equals its length. There is nothing as much and its opposite as that compartment. It is made like a circle, and seems to be a square; a square, and seems to be a circle. The eyes say one thing and the thought another, and meanwhile black bile descends and comes, in place of sight and thought.

In that compartment it is neither hot nor cold nor lukewarm. No wind is there except a dull wind enclosed within a still, dry, and cool wind. And a tall angel whose name is not specified is appointed over the compartment, and the angel does nothing, but his mouth is open like a person whose heart brings up boredom and seeks to yawn.

(S. Y. Agnon, “The Parable and the Moral,” in Ir uMelo’ah, p. 411)

A monk asked Joshu, “Does a dog have Buddha-nature, or not?” Joshu answered, “Mu” (= nothingness, absence).

(“The Dog of Joshu,” the first koan, in The Gateless Gate, a collection of Zen koans from the 13th century)

The three citations above strike us as absurd. They break a dichotomy or a trichotomy: how can a room be neither large, nor medium, nor small? And how can there be a temperature that is not cold, not hot, and not lukewarm? It’s like the compartment fashioned as a circle yet seeming a square. The first two examples are apparently trichotomies, i.e., a division into three exhaustive and mutually exclusive options. The last one is a dichotomy, a division into two such options.

An exhaustive and exclusive partition divides a space into several sets such that every element in the space belongs to one and only one of them (the sets do not intersect). What characterizes such partitions is that they force upon us seemingly unavoidable conclusions. If you are not in set A, then necessarily you are in B (or C). In addition, it cannot be that you belong to none of them. In this column I wish to show that exhaustive and exclusive partitions are a confusing matter, and to bring examples that teach how such partitions can be broken. Some of what follows I have mentioned in the past. Even so, I found it proper to gather these points in one place and lay them within a more general framework for the benefit of the many.

The Three Laws of Thought

It is customary to say that our thinking rests mainly on three fundamental laws: the Law of Identity, the Law of Non-Contradiction, and the Law of the Excluded Middle. The Law of Identity can concern objects or propositions. Regarding objects, it speaks of the persistence of an object’s identity (I am I). One can of course debate what happens when various of my attributes change, and the like. On the logical plane it says that every proposition is equivalent to itself. The Law of Non-Contradiction says that it cannot be that a proposition and its negation are both true at the same time. The Law of the Excluded Middle says that either a proposition is true or it is false; there is no third option.

To the layperson these three (especially the Law of Identity) sound trivial, and some even appear equivalent (though the very notion of equivalence relies on them). But mathematicians and logicians tend to articulate and formulate their axioms and assumptions explicitly, to avoid possible confusions. There are theorems that look utterly trivial whose proofs can take months. The reason is that precise thinking requires explicit formulation of all assumptions laid on the table. For example, the Intermediate Value Theorem(s) state that when a real-valued function that is continuous assumes two different values, it assumes every value in between—does that sound trivial? Indeed, for this is an intuitive property of continuous functions, as functions you can “draw without lifting your pencil from the page.” Even so, mathematicians prove such theorems with great detail and precision.

But if one truly takes pains to formulate these laws, a natural question arises: could they have been false? Are they laws like Newton’s laws of mechanics, or the laws of chemistry? From another angle, we may ask: whence do we derive them? Are they the results of observation, like the laws of nature?

I have often remarked (see, e.g., this article, and many other places on the site) that the term “laws of logic” or “laws of thought” is very misleading, since it reminds us of other kinds of laws, like the laws of physics or of the state. But there is a vast difference between the laws of logic and any other system of laws. The laws of logic are not the result of observation; their truth is built into the nature of things. In modal terms, there may be worlds in which different laws of physics hold, and certainly legal systems with laws different from ours, but there cannot be a world in which the laws of logic differ from ours. A logical law adds nothing beyond the properties of the thing under discussion (it is analytic, not synthetic, and therefore a priori rather than a posteriori).

This is also the answer to the question of their source. There is no need for a legislator or for anyone to confer authority upon them. Their authority stems from themselves. The authority of a law says that it must not be otherwise. The laws of logic have no authority because it cannot be otherwise. Therefore one who asks about the source of these laws does not understand them. They have no source and need no source. Hence, one cannot dispute them (i.e., think otherwise).

Even so, in the history of logic and philosophy there have been many challenges to the Law of the Excluded Middle (it is the most attacked of the three). As we will see, the Law of the Excluded Middle is the logical basis for discussion of dichotomies and trichotomies, and so we now turn to it.

The Law of the Excluded Middle[1]

The Law of the Excluded Middle states that either X is true or it is not true; there is no third option. One might say that the Law of Non-Contradiction tells us that two opposite propositions (X and “not-X”) cannot both be true. The Law of the Excluded Middle states that they cannot both be false (one of the two must be true). This probably sounds similar to you, but they are two different laws. Note that from Non-Contradiction one can derive from the truth of X the falsity of “not-X.” But can one derive from the falsity of X the truth of “not-X”? The answer is no. The Law of Non-Contradiction says nothing about that. It is the Law of the Excluded Middle that yields this (that one of them must be true, since there is no third option). From here it also follows that double negation cancels: from the falsity of “not-X” we derive the truth of X (for it cannot be that both are false). Proof by contradiction is likewise based on the Law of the Excluded Middle (see Column 654): if we proved that “not-X” is not true (e.g., because it leads to a contradiction), we may infer that X is true.

Another formulation that will aid us below says that the Law of Non-Contradiction teaches that X and “not-X” exclude one another, whereas the Law of the Excluded Middle teaches that these two propositions together exhaust all possibilities (see the opening of the column). This means the two laws are different and independent, and thus in principle one can adopt a logic that contains Non-Contradiction (it is needed in any logic whatsoever) but not the Excluded Middle. In such a logic it is not correct to say, for every sentence X, that either it is true or it is not. Hence, if we have denied “not-X,” we cannot infer X, because there is also a third option different from both. Proposals for such logics will be presented later in the column.

The Logic of Dilemmas

We have seen that proof by contradiction rests on the Law of the Excluded Middle. The same holds for dilemma arguments (what the Talmud calls “mah nafshakh”—either way). Consider the following pattern of argument:

X –> P

~X –> P

———-

P

[The symbol ‘~’ denotes negation.]

This argument says that if the assumption X entails P, and the assumption “not-X” also entails P, then we may conclude P (i.e., that P is true). Why? Because by the Law of the Excluded Middle either X is true or “not-X” is true, and there is no third option.

For example, we can assign the following readings to the propositions in this schema: X is the claim that a certain action has a cause, and ~X is the claim that it has no cause. P is that the act was not performed by choice. The first premise says that if the act has a cause then it is not done by choice. The second says that if the act has no cause, then too there is no choice. The conclusion: the act was not done by choice. Why? Because there is no third option: either it was done for a cause or not; either way, there is no choice. Of course, these premises are supposed to hold for any act whatsoever, so the conclusion is general: we have no free will. This is the argument I brought in Column 645 from van Inwagen.

Some Common Challenges to the Excluded Middle

Some philosophers have noted that the Law of the Excluded Middle is relevant only within the semantic field of the subject at hand. For example, the statement that the wind is green or not green has no meaning, because wind has no color. This means that if we look for the wind among green things and do not find it, it does not follow that it is among things of another color, not green. It is not there either, for it has no color. Likewise, if we look for the king of France among the bald and do not find him there, we should not infer that he is hairy. You will not find him among the hairy either, because for the last two hundred and fifty years France has had no king.

Of course, one can rescue statements of the first sort by adopting a different reading of negation (see a similar distinction in Columns 348 and 376). When I say “not green,” I mean “it is not true that it has the color green.” If that is the intent, then the Law of the Excluded Middle applies also to the greenness of the wind. The correct statement is that the wind is not green. If the negation is interpreted as “having some other color,” then indeed neither statement has meaning with respect to wind. But the second example, regarding the king of France, cannot be interpreted thus. It is not true that the king of France is bald, and it is not true that he is not bald (hairy). Admittedly, here too one can say: it is not true that the king of France is bald.

Similarly with the familiar loaded question: “Have you stopped beating your wife?” Apparently the answer is yes or no, with no third option. But if I have no wife, or if I never beat her, we return to the previous challenges. It is not true that I stopped, and not true that I did not stop.

The best-known challenge is due to the Polish logician Łukasiewicz. He argued that for statements about the future it is not correct to apply the Law of the Excluded Middle. He bases this on logical determinism (see Column 301). Logical determinism is an argument that goes like this: first, note that if tomorrow there will be a sea battle, then the proposition “Tomorrow there will be a sea battle” is already true today (for there is a fit between it and the state of affairs in the world it describes, even if we do not yet know it. A proposition whose content fits the state of affairs it describes is true even if no one knows it). Second, if already today it is true that tomorrow there will be a sea battle, then its occurring tomorrow is necessary (there is no possibility that it will not occur). These two premises comprise what is called “logical determinism.” Now add a third premise: by the Law of the Excluded Middle, either there will be a sea battle tomorrow or there will not. It follows that every event that occurs occurs necessarily. There is no choice and no chance in the world at all. This is essentially a dilemma of the sort we encountered above. Finally, Łukasiewicz argues that if one assumes there is choice or chance (or at least that such mechanisms can exist—that is, they are not contradictory), then necessarily the Law of the Excluded Middle cannot be applied to propositions about the future.

One can quibble over why it helps Łukasiewicz to give up the Law of the Excluded Middle. Even without it—i.e., even if we assume there is a third option (what would that be? Perhaps the concept of “sea battle” will have lost its meaning—there will be neither seas nor battles)—it would still be true that what will happen will happen of necessity. If logical determinism is correct, then renouncing the Law of the Excluded Middle does not save choice or chance. But there is no need for this, since the premise of logical determinism is not correct (see that column for why). Thus, this challenge to the Law of the Excluded Middle also falls.

Bottom line: the Law of the Excluded Middle is stable and strong, and there is no substantive refutation of it (at least so long as there are no excursions beyond the relevant semantic field, as above). To my understanding there is no real logical justification to give it up, even if its force is less than that of Non-Contradiction. Still, we shall now see some relevant caveats.

Dichotomous Thinking

We saw that the Law of the Excluded Middle presents us with a picture in which there are only two exhaustive options—that is, no third. From this, people tend to infer logical conclusions in various domains, by two main tools defined above: dilemma arguments and proofs by contradiction. As noted, both rely on the Excluded Middle. Yet in many cases one has the sense that something is amiss. Such arguments are presented to me and I still do not accept the conclusion. Does this mean I reject the Excluded Middle? In light of the above, it is very hard to take refuge in such a logical exit. This law is firm and clear and hard to forgo. This means that if I trust these intuitions, there should be other ways to loosen dichotomies without abandoning the Excluded Middle. I can say for myself that almost always when a dichotomous picture is presented to me, divided into two options, I identify with neither. How can that be reconciled with the Excluded Middle?

In the second part of the column I will present a toolbox for dissolving dichotomies, comprising several types of tools (with certain relations among them). Before I describe the tools, I will present an example on which we can demonstrate them all.

An Accompanying Example: Religious Zionism

There is a common dichotomy in the world of religious identities between Haredi (ultra-Orthodox) and Religious-Zionist. Why is this a dichotomy? Because it is based on the Excluded Middle. Assuming you are a religious person, there are only two options: either you are a Zionist or you are not. There is no third option. Now, if you are not a Zionist you are Haredi, and if you are a Zionist you are Religious-Zionist. Hence there is no other religious identity beyond these two. QED. This looks like a necessary conclusion, and yet I will try to show several ways to attack this argument. I will then conceptualize them and move on to describe the toolbox itself.

The conclusion that there is no other identity is of course too strong. Who said my identity must be based on whether I am a Zionist or not? It is true that necessarily either I am a Zionist or I am not, but my identity need not be Zionist or non-Zionist. I can be identified around acts of kindness to others, promoting the mitzvah of petter chamor (the firstborn donkey) among the masses, standing on one leg every morning, veganism, environmentalism, and so on. This is a simple mistake in moving from the dichotomy to a particular implication of it—identity. But even if we set identity aside, the dichotomy itself appears correct: either you are a Zionist or not. Yet even here the options are not exhausted.

We can define two kinds of opposites to a Zionist outlook: anti-Zionism (a contrary opposite) and a-Zionism (a nullifying opposite). Thus, even if we address Zionism apart from identity, the dichotomy is not exhaustive. In addition, I have often noted the matter of the hyphen. Yosef Burg (a leader of the NRP, former minister, and Avrum [Avraham] Burg’s father) once said that the essence of Religious-Zionism is neither the Zionism nor the religiosity, but the hyphen. One who defines himself as Religious-Zionist has Zionism that is religious and religiosity that is Zionist. From this we see there is another group that escaped our notice: religious people who are also Zionists but without the hyphen. Their Zionism is secular, not religious. They are Zionists just like any secular Israeli. It does not flow from religious motives, and they do not necessarily see the State of Israel as the dawn of redemption. They simply want to live here, whether because of the commandment to settle the land, or simply because they are fed up with the gentiles and want to live among their own people, just like Belgians or Tanzanians.

On this matter there is a famous story about Yeshayahu Leibowitz, who was asked by a group of foreign journalists going around interviewing intellectuals why he was a Zionist. Leibowitz answered: because I’m fed up with the gentiles. Add to this the story about the Ponovezh Rav (Rabbi Kahaneman, who founded the Ponovezh Yeshiva in Bnei Brak), who would neither say Hallel nor omit Tachanun on Independence Day. People challenged him about his consistency: if he is Haredi he should say Tachanun, and if he is Religious-Zionist (heaven forfend) he should say Hallel. The Rav answered that he is a Zionist like Ben-Gurion. Ben-Gurion too did not say Hallel nor omit Tachanun on Independence Day (nor on any other day). Haredim usually take this story as a joke at the expense of foolish Zionists, but clearly they themselves are the foolish ones. The Ponovezh Rav was not telling a joke. It was his way of declaring that he was a secular Zionist. Here is another option alongside Religious-Zionist (with the hyphen) and Haredi.

Another way to challenge this dichotomy is of course to say: I am not religious at all (or even a non-Jew). Therefore I am neither Religious-Zionist nor Haredi. Or simply: I have no position on these matters. In short, I am not in this game.

I now turn to the conceptualizations. I will describe the general toolbox for dissolving dichotomies, and the example of Religious-Zionism and the alternatives to it will serve to illustrate the matter.

First Tool: Nullifying vs. Contrary Negation

Since dichotomies are always built on negation, the first tool we must examine touches negation itself. I noted that there are two kinds of logical negation: nullifying and contrary. 0 is the opposite of 1, but so is −1. The first is a nullifying opposite and the second is a contrary opposite. For example: darkness is a nullifying opposite of light, but cold is a contrary opposite of heat. How do I know? Because adding cold and heat offsets and yields something lukewarm; but adding light to darkness leaves us with light. Thus the model for heat-cold is 1 vs. −1 (when you add them they cancel), whereas the model for light-darkness is 1 vs. 0 (adding them leaves the 1). I hinted at this distinction earlier in the section on challenges to the Excluded Middle (and pointed to Columns 348 and 376).

Thus, in any dichotomy presented to us, we should examine whether it is based on contrary negation or nullifying negation. If the negation is contrary, that already hints at a direction for a third option: the nullifying negation. In the case of Religious-Zionist versus Haredi, if the negation is contrary, it sets Zionism against anti-Zionism, but that leaves us the a-Zionist option.

In this way we move from a dichotomy to a trichotomy—from a picture of two exclusive options to one of three. Below we will see more tools that move us to trichotomies, and tools that help to dissolve trichotomies as well.

Second Tool: Splitting One Side in Two

In quite a few cases the dichotomy presented to us is indeed correct, but one of its sides assumes a non-necessary premise. In such a case we can show that one side of the dichotomy splits into two different options, thereby solving the problem. I will bring two examples.

The first is again Religious-Zionism. We saw that the main component in Religious-Zionism is the hyphen. From this it follows that even if a person is both Zionist and religious, there are still two possibilities: Religious-Zionism with a hyphen (a person whose Zionism is religious and whose religiosity is Zionist) and one who is religious and Zionist without the hyphen (a religious person whose Zionism is secular like Ben-Gurion’s).

The second is a common argument against free will. In Column 645 I brought an argument against libertarianism (belief in free will) based on a dichotomous premise: an act either has a cause or it has no cause. There is no third option. If it has a cause, it is deterministic (for a cause is a sufficient condition: when the cause occurs, so does the effect). If it has no cause, it is random (indeterministic). Either way, there is no choice. The structure is exactly the dilemma pattern described above (we prove there is no choice, P, by showing that the conclusion follows from both horns of the dilemma).

Again, there is a mistake, because under the heading of indeterminism (an action without a cause) two different mechanisms can appear: randomness and choice. The difference is that choice has a purpose (a person who chooses acts to realize a purpose or value) whereas randomness does not. That is, the dichotomy is correct: either there is a cause or there is not. But the “no cause” horn comprises two different options under it. See there for details.

This method too yields a trichotomy, since one horn of the dilemma splits into two.

Third Tool: Moving to Implications

We saw above, in the Religious-Zionist example, that even if we accept the dichotomous picture regarding your worldview—either you are a religious Zionist or a religious non-Zionist—sometimes in moving to the conclusion we take one more step, and it too deserves attention. In the case we are discussing—religious identity—it may be that a person will say: true, I am a religious Zionist, but that is not my identity. There is another value that is primary for me, and it defines my identity.

I gave an example of this in my manifesto (Column 500). There I explained that the heading “Haredism” hides under it two different, almost independent ideas: opposition to modernity and opposition to Zionism. Therefore I would expect that opposite Haredism would stand two different groups: modern religious and religious-Zionist. In practice, in religious discourse there is opposite Haredism only one group: Religious-Zionism. Already here it is clear that one can define one’s identity around modernity rather than around Zionism, and thereby produce a third identity different from Haredism and Religious-Zionism (a person who identifies as a modern religious Jew can of course also be a Zionist. But his Zionism is not perceived by him as the core component of his identity). Beyond that, I explained there that this is also the right thing to do, because the debate over Zionism has lost most of its meaning. It is an anachronism that remains with us only due to inertia. The debates raging today revolve almost entirely around modernity, not Zionism. Either way, this is an example where, when we move from the dichotomy to its implications, we sometimes discover additional options. Even if the dichotomy as such is correct, it does not mean we may base identity upon it.

Fourth Tool: Attacking a Shared Assumption

We have already encountered this tool above. The assumption is that whenever there are two opposites, there is always something in the background common to them both, and that can be attacked (or disputed). “Salty” is not the opposite of “triangle,” because they do not concern the same kind of thing: salty is a taste and triangle is a geometric shape. Salty is the opposite of sweet because both are tastes. That is, if a dichotomy of two opposing options is presented, there will always be in the background an additional assumption common to the two views on either side.

Look again at the Religious-Zionist example. What is common to Religious-Zionism and Haredism (= religious but non-Zionist)? Both assume that reality must have a religious meaning—that history must have religious significance. Either Zionism and the State of Israel are the dawn of redemption, or they are the work of Satan. But that shared assumption itself can (and should) be attacked: who says there is a meaning or significance to every historical event or movement? The world runs its course, and so it is not true that Zionism is a movement driven by the Holy One, blessed be He, nor that it is driven by the sitra achra. It is simply a movement of people seeking to realize an idea. One may agree or disagree, but this does not require us to go behind the metaphysical-theological curtain and speculate about the spiritual force underlying it. Perhaps there is no such force at all.

This move is very similar to “removing the hyphen” we saw above. Here too a third option is produced, except that this time it is an attack on an assumption common to the two sides of the dichotomy, not a splitting of one side into two different species. I think one can do this in other cases as well. For example, in the question of free will, what is common to randomness and determinism is the absence of purpose. There is a view that there are no actions in the world directed toward any purpose. The third option points out that this common assumption itself may be false, and thus a third option arises. Admittedly, with respect to free will the connection to the previous tool seems more artificial than in the Religious-Zionist example.

Another way to attack a hidden assumption common to both horns of a dilemma appears in the example above regarding the king of France. There too the shared assumption was that France has a king; now the question is whether he is hairy or bald. But if there is no current king of France, he will not be found on either horn of the dilemma. De facto, a third option has arisen. So too with Religious-Zionism: if the person in question is not religious at all, he does not enter the dilemma (religious-Zionist or religious non-Zionist). If one has no sister, one need not decide whether she is promiscuous or not.

Returning to nullifying and contrary negation: I illustrated it via the relation between the number 1 and two other numbers. −1 is its contrary negation and 0 is its nullifying negation. But this follows from a modern view of 0 as if it were an ordinary number whose value lies exactly between 1 and −1. One can also regard 0 as not a number at all, for it does not describe a quantity but an absence (see a similar distinction in the article Midah Tovah for Parashat Balak 5767, cited also here. See also Column 340, the section on “tzimtzum literally”). Along similar lines wrote Rabbi Zeini, in his article in Sefer Higayon, about the empty set: that this is a formal definition, but in ordinary meaning it is not a set at all.

Note that 0 and the empty set here play the role of an attack on a shared assumption. When I discuss the relation between 1 and −1, both of which are numbers, I introduce an option that is not a number at all (and not merely a number that lies in the middle).

We can see that this tool too creates a trichotomy. But it also allows us to dissolve trichotomies, since it can attack the common axis. We do not generate a third option but rather reject both options and move to a different plane of discussion (from numbers to what is not a number).

Fifth Tool – Pointing to Intermediate States: Fuzzy Logic

We saw that the Excluded Middle expresses dichotomous thinking: there are only two ways to treat any proposition—true or not. The last tool I will present for dissolving dichotomies is to move to fuzzy logic, i.e., a logic in which there are more than two possible answers, usually a continuum (continuous logic).

The classic example is the Sorites (heap) paradox (see Column 110 and elsewhere). The paradox is built from three claims each of which seems quite reasonable: (a) one pebble is not a heap; (b) adding one pebble to a given pile does not change its status; (c) a thousand pebbles constitute a heap. As noted, though each of the three claims seems very plausible, the three do not cohere. If adding one pebble at a time never changes the pile’s status, there is no way to get from one to a thousand such that the thousand are a heap.

The solution to this paradox (and many equivalent ones) is to give up the dichotomous conception of concepts like “heap.” Premise (b) must be replaced with the assumption that adding one pebble changes the status a bit—that is, slightly increases the degree of heap-ness of the pile. The assumption is that piles of pebbles are not divided in a binary way into heap and non-heap, but lie on a continuum of degrees of heap-ness (we can describe it by levels of heap-ness between 0, not a heap at all, and 1, a full heap; in between are various degrees, 0.3, 0.74, 0.91, and so on).

This means that sometimes a dichotomy rests on an incorrect logical assumption. Our everyday concepts cannot be examined through the prism of binary logic—yes/no, black/white—but through continuous/fuzzy logic. This of course adds countless options to the two posed by the original dichotomy. A pile can be a heap or not a heap, but it can also have a degree of heap-ness of 0.3 or 0.77. This is another way to break dichotomies and make them polytomies.

Of course, the previous tool—attacking a shared assumption—can dissolve polytomies as well. If there is something that does not play on the field of heap and heap-ness at all, then we have dissolved the entire polytomy.

Back to Agnon

If we return to the Agnon statements that serve as the column’s motto: can there be a room that is not small, not medium, and not large? Yes. If it is a room on the internet, for example, that does not lie within the plane of discussion of physical size. There can also be a room whose size is at some level that cannot be defined as small or medium or large. Likewise there can be a place that is not hot, not cold, and not lukewarm—a place with no temperature at all (e.g., an internet site). These two are examples of breaking trichotomies, which is possible if one claims that the plane of discussion is not relevant to our subject and moves to a different plane.

In the Zen story there, the picture is even more called for. Who says the dog under discussion can be measured in terms of having or not having Buddha-nature? The term “Mu” used in the monk’s answer denotes absence or lack. It is precisely a nullifying negation of the two answers. The third option there arises by attacking the axis along which the dichotomy is stretched.

[1] On this law and the challenges to it, see Hugo Bergmann, Introduction to Logic, p. 245ff.