A Look at 'Begging the question' (Column 732)

With God's help

In almost every logic textbook you’ll find a list of logical fallacies. Among them, in a place of honor, you’ll usually find ‘begging the question’. A quick search here on the site will also often turn up appeals to it, on the assumption that it’s a fallacy. Yet in a past article of mine I explained that begging the question characterizes every valid logical argument, and therefore it’s a mistake to place it in a list of fallacies. Only certain very specific instances of question-begging might perhaps be seen as fallacious, and even that isn’t precise. More than once in the past (see for example here) a debate on this arose on the site, and so I thought it appropriate to devote a column that would present the issue systematically.

The Three Fundamental Modes of Inference

In logic we commonly distinguish among three modes of inference: deduction infers from the general to the particular; analogy infers from one particular to another; and induction infers from the particular to the general. For example, if all humans are mortal and Socrates is a human, then Socrates is mortal. This is a deduction, since one of the premises is general (about all humans) and the conclusion is about one individual from that class. This is, of course, a necessary inference, since one cannot accept both premises and reject its conclusion. In contrast, the argument “Moses is mortal; therefore Socrates is also mortal” is an analogy. It infers from one particular to another that is similar in some respect. This is obviously a non-necessary inference, since an analogy can turn out to be false (that happens when the similarity between the source and the target is irrelevant). An inference that starts from the premise that Moses is mortal and concludes that all humans are mortal is an induction. It takes a particular and infers from it a conclusion about the entire class (all items similar to the initial particular). Clearly this, too, is not a necessary inference.

The Difference Between the Types of Inference: The Necessity of Deduction

I have explained many times the root difference between these inferences. Why is only the first necessary while the other two are not? The first adds no new information. The information found in the conclusion (“Socrates is mortal”) was already embedded in the premises (“all humans are mortal,” and “Socrates is a human”). The other two modes add information beyond what is contained in the premises. The conclusion that Socrates is mortal contains information not found in the premise that Moses is mortal; and of course the conclusion that all humans are mortal is not found in the premise that Moses is mortal. Inferences that add information beyond what is embedded in their premises are not necessary.

Needless to say, one can complete non-necessary arguments and turn them into necessary ones (this is sometimes called “completing enthymemes.” An enthymeme is an unstated premise; see more below). For example: Moses is mortal, and what is true of one person is true of all humans; therefore all humans are mortal. This is already a deduction, since if one adopts both premises, the conclusion indeed follows of necessity. This was with respect to induction, but the same holds for analogy: from those same two premises one can also derive the conclusion that Socrates is mortal. This is a weaker conclusion, but of course those two premises also contain it.

From here it’s easy to understand the secret of deduction’s necessity. Why must one who adopts both premises also accept the conclusion? Because it is contained within them. If you’ve adopted the premises, then implicitly you’ve adopted the conclusion. If you concede that all humans are mortal and that Socrates is a human, then within those two premises you have already conceded that Socrates is mortal. This didn’t first arise in the conclusion; it was already embedded in the premises. You can see this more easily if you unpack the general premise “all humans are mortal,” and note that it is actually a shorthand for countless particular premises: Moses is mortal, Jacob is mortal, Ahmad is mortal, and so on—and of course Socrates (who is also a human, per the second premise) is mortal.

The conclusion is that the purpose of a deductive argument is not to reveal something new but to expose to us information that is embedded in the premises—nothing more. Analogy and induction are inferences that add information beyond what is embedded in the premises, and precisely for that reason these are not necessary patterns of argument.

It is important to understand that analogy and induction are not necessary patterns of argument, but that doesn’t mean it’s wrong to use them. These are important tools we all employ. Still, if we examine an argument presented as deductive and discover that it is actually an induction or an analogy, then indeed we have found a fallacy. A fallacy in a logical argument means that the conclusion does not follow of necessity from the premises—and that is certainly true of analogy and induction: their conclusions do not follow of necessity from their premises. More generally, the fact that there is a fallacy in an argument does not mean it is worthless, and certainly not that its conclusion is false. It only means that the conclusion does not follow of necessity from the premises (see an example of this in my article in Akdamot 11).

A Note on Completing Enthymemes

In everyday speech a person does not always state all of their premises, relying on certain things being self-evident. Therefore, the mere fact that someone’s argument contains an implicit, unstated premise is not a fallacy. In everyday speech one may shorten and omit things that seem self-evident. In mathematics and logic we tend to be more punctilious, though even there you’ll often find implicit assumptions. We saw above that an argument by analogy is actually based on a tacit premise that is never placed on the table: that what is true of particular X is true of particular Y that resembles it. The same is true of induction (that what is true of particular X is true of the entire class of particulars similar to it).

For example, you can hear arguments for changes in halakhah built roughly as follows (see in the series of columns 475 – 480): in the Talmud women are disqualified from testimony; today women are educated and involved in social and economic life, and therefore they should now be qualified to testify. Without entering into the premises and argument—and not into the conclusion—note that there is a logical leap here. There is a tacit premise according to which the Talmudic disqualification of women’s testimony stemmed from the fact that they were uneducated and uninvolved in social and economic life. To the person presenting the argument this may sound self-evident, but once it’s placed on the table one can wonder whether this is indeed so obvious. You’ll see that the discussion can look entirely different once the missing premises are completed.

Therefore, it’s a good habit to complete the enthymemes (as mentioned above), that is, explicitly place on the table the argument’s implicit premises before beginning to discuss it. This can reveal surprising findings (those premises are not always as self-evident as one thinks).

Is Begging the Question a Fallacy?

In light of what I have described so far, it is not clear why begging the question should be a fallacy at all. Consider the common argument: all humans are mortal; Socrates is a human; therefore Socrates is mortal. We saw that the conclusion is embedded in the two premises and does not add anything beyond them, and precisely for this reason the argument is valid. If that were not the case, there would be no valid argument here (as with induction and analogy). So in fact this argument begs the question, for the conclusion is contained within its premises. But by all accounts this is not a defective argument. On the contrary, a valid logical argument is considered the antithesis of a faulty one. Note that we have reached a picture in which we can formulate the following argument: every valid logical argument is built on begging the question (that is why it is valid). A valid logical argument is not fallacious. Conclusion: begging the question is not a fallacy.

Let us take a more blatant example of begging the question: whence that every Jew must wear a hat? Because it is written “And Abraham went,” and a Jew like him didn’t go without a hat. If Abraham went with a hat, then all of us, his descendants who are required to follow his ways, must wear a hat. Q.E.D. Here, I assume everyone chuckles at the absurdity of this argument. It begs the question, for behind the premise that “a Jew like him didn’t go without a hat” lurks, implicitly, the conclusion that every Jew must wear a hat. That is, the very conclusion we sought to prove was one of the premises in the argument proving it. This is blatant question-begging, and this really seems like a fallacy.

Well, it is not a fallacy. An argument that assumes X and proves from it X is a perfectly valid argument. If one assumes X, one can indeed infer X. It is impossible to adopt that argument’s premise and reject its conclusion; as noted, that is the definition of a valid argument.

The Question of Persuasive Value

One may, of course, wonder about the value of such an argument. What is the point of presenting it? Consider Reuven, who claims that X is false, and Shimon comes to prove to him that X is true. He presents the following argument: X, therefore X. The argument is certainly valid, since whoever adopts the premise must adopt the conclusion. But Reuven will not adopt the premise (for that is exactly the claim he is disputing with Shimon). In other words, the argument is valid, but it lacks persuasive value. There is no point in using it to persuade someone who rejects the conclusion, since he will not accept the argument’s premise and therefore will not be persuaded by its conclusion. Note that validity concerns only the hypothetical derivation of the conclusion from the premises. A person can agree that the argument is valid, but if he does not accept its premises he will not accept its conclusion either.

In other words, unlike other logical fallacies that concern problems in an argument’s logical derivation, begging the question is unrelated to that. The logical derivation in an argument that begs the question is perfectly kosher; that is, the argument is valid. The problem is that it is unhelpful because it rests on a premise not accepted by the interlocutor (the argument’s addressee).

What about the Socrates argument? As noted, it is valid, since its conclusion follows of necessity from its premises. Is it useful? Sometimes yes. Since the conclusion does not appear as one of the premises but is only contained in their combination (you need both the premise that all humans are mortal and the premise that Socrates is a human to arrive at the conclusion), there may be value in the argument, for sometimes a person who agrees to both premises errs and denies the conclusion. The argument shows him his mistake and indicates that if he concedes both premises, he must also adopt the conclusion. In the Socrates argument this still sounds too simple, so it also seems fairly valueless; but in more complex logical and mathematical arguments, this can very much be the case.

Consider, for example, Euclidean plane geometry. Take a teenager and explain to him its axioms (through two points there passes only one straight line; two parallels do not meet; etc.). Now ask him: what is the sum of the angles in a triangle? A typical teenager will not be able to answer this question, although after he studies geometry he will see that from the premises he knew and understood well in advance he could have derived the answer: 180 degrees. The reason is that deriving this conclusion from the premises of geometry is complex and involved, and therefore an ordinary person will not succeed in doing so without a teacher’s help. In such a case, although the conclusion follows logically from the premises, and in the sense described above it is contained in them, the mathematical proof of the triangle-angle-sum theorem is a valid logical argument—and as such it begs the question. Still, it cannot be said that the proof has no value. It has value because it exposes to us information embedded in premises we accept, but of which we were not necessarily aware. The proof revealed it to us; that is, it had value. Such a proof is an argument that begs the question, but it is not correct that it is worthless. Malcolm once said that some claim is indeed a tautology, but a valuable tautology (it is hard to see its conclusion until it has been shown to us). I have often written that the evolutionary thesis of “survival of the fittest” is a tautology, but see how many years passed before it was noticed, and how many fascinating and novel conclusions have been derived from it. It is a valuable tautology.

Back to Begging the Question

If so, begging the question is not a fallacy in the sense that there is some logical defect in the argument. Its conclusion does indeed follow of necessity from its premises. At most one can speak of the lack of persuasive value of such an argument. But persuasive value is not a black-and-white matter. Very intelligent people may discern a conclusion from a set of premises that less intelligent people will fail to draw. For the former, the argument will lack persuasive value; for the latter, it may indeed have persuasive value.

An argument whose conclusion appears as is among its premises (premise: X; conclusion: X) is always devoid of persuasive value, and therefore it is customary to regard it as a fallacy. But if the conclusion requires analysis and a complex logical derivation from the premises, then it may have great value. As we have seen, all mathematical theorems are of this sort. Mathematical proofs are valid logical arguments whose derivations are far from trivial, and therefore they have value. In contrast, the laws of nature that science arrives at are based on analogy and induction, since science deals with collecting data and generalizing them into universal laws. Therefore, its arguments are not valid logical arguments.

An Exercise in Begging the Question

Before what follows, a warm recommendation to watch videos in Yosef Lavran’s Chachmoblog series. He has a collection of many fairly short videos (about a quarter of an hour), and many of those I’ve heard (mainly on walks with my dog Peanut—may she live and be well—whose merit should protect us) were excellent. Truly a masterpiece of systematic and honest thinking in various domains, presented very clearly with good examples. I did find inaccuracies there from time to time, but they are negligible. Highly recommended.

This morning I watched a video of his dealing with the basics of logic and the identification of bad arguments. Toward its end (around 13:20) he presents an argument that begs the question, which pricked up my logical antennae:

Premise: The Torah says that the Torah is the word of God.

Conclusion A: Therefore the Torah is the word of God.

Conclusion B: Therefore everything written in the Torah is true.

Between the two conclusions there is of course a logical leap, for there lurks an additional premise: that the word of God is true. But that is a standard completion of an enthymeme, and certainly not a fallacy. I want to focus here on the step from the premise to the first conclusion. Lavran argues that it begs the question, since that step assumes that everything written in the Torah is true. But that is precisely the conclusion he wished to prove.

On further thought, however, that is not exactly the case. To move from the premise to conclusion A, one need not assume that everything in the Torah is true, but only that this particular statement (that what is written in it is the word of God) is true. For example, one could base this on the premise that all factual claims in the Torah are true; therefore, if the Torah states that it is the word of God, then apparently God said so. Now we can conclude that everything written in it, including what goes beyond factual claims (such as commandments), is true, because the word of God is always true (true in a broad sense: including that it is normatively binding). Alternatively, one could base it on the premise that everything written in the Book of Exodus is true, and from there infer that everything written in the Torah is true. Here, too, this is not begging the question, since the (implicit) premise is not identical with the argument’s final conclusion.

Seen this way, the argument Lavran presented does not beg the question at all. What it requires is only a simple completion of an enthymeme, and as we have seen, that is not a fallacy. Of course there is no necessity to accept the premise that the Torah’s factual claims are true, but that is true of every logical argument: every argument is based on premises, and naturally the listener must accept the premises to be persuaded of the conclusion. There is nothing special about this argument. It is no worse than any other logical argument (aside from the fact that its formulation requires completion, as is common in everyday language), which addresses those who accept its premises.

Indeed, there are several ways to complete the missing enthymeme here, and the presenter did not hint which one he had in mind. Some completions are more reasonable than others, and one of them is the path we assumed at the start: that everything the Torah says is true. If that is the missing premise, then indeed this is begging the question—but as we have seen, there is no necessity to complete it that way. Here the principle of charity comes into play, instructing us to complete the argument in the most generous way to the arguer and only then discuss it (see on this in column 440). If we proceed this way, we get a perfectly reasonable argument with no begging the question at all. Admittedly, it’s not certain that the arguer intended this, but it’s not certain that he didn’t, either. This is another drawback of an imprecisely formulated argument: it allows for several modes of completion. The arguer can count on our fairness and creativity to complete the argument for him—perhaps better than he himself intended originally.

Is This an Induction?

From this perspective it might seem that adding the enthymeme only shows that this argument is not a deduction but an induction: from the premise that the Torah’s factual claims are true, we infer the broader conclusion that everything written in it is true. But induction is indeed a fallacious logical argument, since as we saw above, if it’s an induction then the conclusion does not follow of necessity from the premises.

But that is a mistake. This is a full-blown deduction. We are not generalizing from factual claims to all claims; rather, we are passing through the premise about the reliability of the word of God. This argument shows that if one adopts the premise that the Torah’s factual claims are true, the conclusion that all its claims are true follows from it of necessity and not by induction alone. This is a decidedly non-trivial conclusion, and therefore beyond the fact that the argument is valid (and as such always begs the question), it also has persuasive value. In short, contrary to the initial impression, this is an excellent argument.

Back to Abraham and the Hat: Begging the Question and the Principle of Charity

But now I returned to the “Abraham and the hat” argument, which has accompanied me for many years as a paradigmatic example of begging the question. I saw that even there one can perform the same analysis and discover that it is not necessarily begging the question. Note the logical step in question: it is written “And Abraham went,” and a Jew like him did not go without a hat. Why would a Jew like him not go without a hat? Not necessarily because every Jew must wear a hat (in which case this is indeed begging the question), but because a Jew like him would not go without a hat. What is the relevant classification? What does “like him” mean? Various interpretations can be suggested: an elderly Jew, an important Jew, the father of the nation, and more. If so, even here it is not necessarily begging the question. After completing the enthymemes, this too can be seen as an ordinary valid logical argument.

It does seem, however, that here it is harder to find good candidates for that implicit premise. In the argument about the Torah that Yosef Lavran presented, we saw that there were a few decent candidates, and therefore it is reasonable there to see a good argument with no begging the question. Here, by contrast, the completion is less natural, and therefore it is still more plausible that this really was a case of begging the question (i.e., that the implicit premise the arguer intended was that every Jew must wear a hat). Even the principle of charity has its limits (see more on this in this article and in column 30).