Another Look at the Connection Between Logic and Facts (Column 752)

With God's help

Dedicated to my former friend, Rabbi Raphael Israel Maayan, may he live and be well

If you resolved one of R. Akiva Eiger's unresolved difficulties, you probably made an even number of mistakes

(Rafi Maayan)

A few days ago I saw a YouTube video in which the speaker presented a logical proof that either you or I do not exist. This is a nice example of problematic use of logical formalization, a topic I have addressed more than once. The formalization leads us to a conclusion that appears absurd on its face, but the route seems logically valid; that is, the conclusion appears, ostensibly, to be necessary. I would like to examine this argument a bit, and use it to sharpen a few important points about the relation between logic and the world.

The Argument

The speaker presents a structure with three claims:

You (the students) exist. I (the lecturer) exist. At least one of the three claims in this set is false.

Claim 1 sounds very plausible to us. The same is true of claim 2. So we shall set them aside (notice: I am not assuming that they are true, only setting them aside). This invites us to begin the discussion with claim 3. Is it true or not? Here we have only two possibilities: a. Claim 3 is false. But then it follows that all three claims in the set are true, including the third. Yet that contradicts the assumption that it itself is false. We have therefore reached a contradiction. So we have proved by reductio that the third claim is true (because the assumption that it is false led us to a contradiction). But if it is true, then at least one of the first two claims must be false, meaning that either you do not exist, or I do not exist, or none of us exists. QED.

Initial Analysis

First, I will ask: what are the premises of this argument? Notice that in the set above no argument actually appears at all. This is not a structure that assumes premises and derives a conclusion from them, but simply a collection of three claims. Nor am I assuming that the structure is correct, that is, that all three claims are true. That itself is the subject of the discussion. At the initial stage I am simply placing this structure before our eyes, and only from here does the argument begin. So where is the argument itself? It begins with an analysis of claim 3. After rejecting the possibility that it is false (because that leads to a contradiction), it reaches the conclusion that it must be true. But if it is true, then its content tells us that at least one of the first two is false (for it itself is true, and there must be at least one of the three that is not). And that is the conclusion of the argument.

So the argument does not appear in the set, but in what is said about the set. Try to think whether you can identify any premises underlying the argument described here. Ostensibly it has none. The claims in the set are the material with which the argument deals, but they are not its premises (an argument does not deal with its own premises; it derives conclusions from them). Kant called such arguments "ontological arguments," and in the past I discussed them in various contexts (for example around the cogito in column 363, and around Anselm's ontological proof—in my book Ha-Matzuy HaRishon, in the first discussion. See also on such arguments in columns 160, 364 and 580).

I will now present two difficulties with this argument.

 

The First Difficulty: Conclusions about the World without Premises

The first difficulty concerns the very formal move itself: how does this hocus-pocus happen, that we arrive at a conclusion without assuming any premise at all? Notice that this is a conclusion about a fact in the world (the nonexistence of people in it). Can one reach a factual conclusion about the world without assuming any premise?

It is true that sometimes one can reach conclusions about the world even without premises. Take, for example, the conclusion that no round triangle exists in the world, or the conclusion that it is not true that the chair beside me both exists and does not exist at the same time. But these two conclusions are not really factual claims about the world. They are the result of a formal logical rule. My information about the world has not been enriched by these determinations. By contrast, the claim that I do not exist, or the claim that you do not exist, or their combination—all these are straightforward factual claims about the world. This is the anti-cogito: I think, therefore I do not exist.

This difficulty was formulated by Kant with respect to ontological arguments in general. He was not prepared to accept that conceptual analysis without factual premises could yield conclusions that are factual propositions about the world. I have already noted in the past that this claim, in and of itself, is not decisive. If there is a valid argument that yields such a conclusion, it only proves that Kant was wrong. The problem here is that the conclusion itself seems unreasonable to us at the intuitive level, which brings us to the second difficulty.

The Second Difficulty: Absurd yet Necessary Claims

The second question about this argument concerns the content of the conclusion: it is patently illogical, since it is clear to us that we all exist. True, that in itself would not trouble me, because the conclusion of a valid argument can be false if not all of its premises are true. Consider the following argument: All frogs have wings. The chair beside me is a frog. Conclusion: the chair beside me has wings. This argument is valid, and the conclusion is silly. That should not trouble us because the premises on which it is based are silly, so it is no wonder that a silly conclusion is derived from them. An argument that derives a conclusion from premises does not say that the conclusion is true. It only shows that whoever accepts the premises must also accept the conclusion. And from this it follows that whoever does not accept them is exempt from the punishment of the conclusion as well.

The problem is that in our argument here there are no premises, and therefore ostensibly the conclusion is necessary (it assumes nothing, that is, it is true under any system of premises we might choose). If so, a further difficulty now arises: is it really true that none of us exists? An absurd conclusion of an argument without premises is doubly troubling.

Whereas the previous difficulty concerned all ontological arguments and not specifically this one, this difficulty arises only with respect to an argument like this. There are ontological arguments whose conclusion is plausible and sensible, such as Descartes' cogito. There the conclusion is that I do exist, and so I have no problem with it on the level of content. With regard to the cogito there is only the first problem: how can one prove a factual conclusion (even if it is plausible) on the basis of conceptual analysis without premises? But here we are dealing with an argument whose conclusion is absurd on its face. Is a valid logical argument enough, even if we have not found a flaw in it, to compel us to accept a blatantly unreasonable conclusion?

Notice that this problem too is not decisive. In every paradox we face the dilemma whether to accept its implausible conclusion, or to assume that there is a mistake in the argument even if we have not found it (see on this in columns 601 and 654). In fact, here too we have a paradox with a similar structure: a necessary logical argument that appears flawless, whose conclusion is absurd on its face. True, here it is an argument without premises, whereas other paradoxes are based on premises (which seem true), but the principle remains the same: in both cases, if we do not want to remain with the conclusion, we must look for the flaw in the argument (and if it is not ontological, then one may look for a flaw in the premises). And what if we do not find such a flaw? As I explained in those two columns, there is the possibility of assuming that there is a flaw but we have not found it (because our logical capacities are imperfect). Thus, for example, in our case, the conclusion that I or you do not exist sounds so preposterous that any reasonable person understands that there is some flaw in the argument. All that remains is to find it.

So we have two difficulties: a. How is it possible at all that we get a necessary conclusion that is a factual claim about the world from an argument without premises? Are ontological arguments possible? b. How should we relate to an absurd conclusion that is derived from an argument without premises (that is, how should we relate to an absurdity whose truth is necessary)?

The Connection to the Liar Paradox

In fact, we have here a version of the liar paradox (see on it in column 195). Consider, for example, this version of it:

• Sentence (a) is false.

This is a sentence that says of itself that it is false. Hence, if it is true, then its content is true, and its content says that it itself is false. But if it is false, then its content is false, and that means the sentence is true. We arrive at contradictions under each of the two possibilities.

I have often explained that self-reference in itself is not problematic. There are claims that contain self-reference and they do not raise any problem, and certainly do not create a paradox. Take, for example, the following claim: every sentence is made of words. This is a claim that also refers to itself, and yet there is no logical problem here. The problem is that in our argument there is self-reference that creates a problem, and in fact, as we saw, two problems. The conclusion that follows from this is that a claim of this kind does not really assert anything. It is a combination of words that has no meaning. We view it as a paradox because of the assumption that it has meaning, but it does not. If a given claim leads to absurdity, one may view it as a combination of words of the type "virtue is triangular," or "every table stands transparently." All these are combinations of words that ostensibly look like claims, but that is only an appearance. These combinations assert nothing.

Unlike the argument we are dealing with here, the liar paradox is not a proof by contradiction but an endless loop. In the case of the liar paradox, neither of the two possibilities (that the sentence under discussion is true or that it is false) is correct, because each of them leads to a contradiction. In our case, by contrast, only one possibility (that claim 3 is false) leads to a contradiction, and therefore this is not a paradox but rather a proof by contradiction that the second possibility is true (that is, that claim 3 is true).

Nevertheless, there is still self-reference here that very much resembles the liar paradox. After all, claim 3 says that at least one of the three claims (of which it itself is one) is false. In fact, what we have here is a sentence of the following type:

(x) The claim {x _* y _* z} is false.    (the asterisk represents "and")

Notice that claim x itself is part of the content of claim x, and the claim is that the combination that includes it is false. This is exactly the structure of the liar paradox as presented above. Except that here we attached two more claims to that claim, and the falsity claim concerns the conjunction "and" of the three of them. If I decompose this claim according to the rules of logic, I get that claim (x) is an "or" combination of three claims:

(x) Either claim (x) is false, or claim (y) is false, or claim (z) is false.

This already looks very similar to the liar paradox, since the first component is exactly the paradoxical sentence that generates the liar paradox. So of course our suspicion is aroused that something smells logically fishy here.

But notice that unlike the liar paradox, if we want to claim here that this claim is meaningless, that would be an ad hoc claim. Merely because the conclusion embarrasses us, we prefer to say that this claim (the third one) has no meaning. To understand the problem, notice that in the liar paradox this is not an ad hoc claim. There, the assumption that this sentence has meaning leads to a logical contradiction, that is, we proved that the claim is neither true nor false. Therefore, in that case, the assertion that this is a paradox is not added ad hoc merely because of our discomfort, but is logically proved. But in our case, the assumption that claim 3 has meaning leads to an embarrassing conclusion (but not to a contradiction). Is it not proper in such a situation to be honest, admit the result, and adopt the embarrassing conclusion? Ostensibly this argument proved it logically.

A Good Reason to Assume that Claim 3 Is Meaningless

In principle, one can attach any collection of claims to one another and create a structure from which any conclusion whatsoever may be inferred. Suppose you want to prove claim A. No problem; all you have to do is build the following structure:

~A     (not A) At least one of the two claims in the set is false.

Now let us repeat the argument from above: suppose 2 is false, then it follows that both claims are true, and therefore clearly 2 too is true, and we have reached a contradiction. From this it is clear that 2 is true (because the assumption that it is false leads to a contradiction), but from its content it follows that ~A cannot be true, that is, A is true. QED. Here you have a machine for proving any claim on earth. Whatever you put in place of A can be proved in this way. For example, you can place there the claim "God exists" and prove it this way, but also the claim "God does not exist" and prove it in the same manner. One can even prove that this structure does not prove A, if we put in place of A the claim "this structure proves A."

What does this mean? Here I have already shown that this structure should not be taken too seriously, since it can serve us for any purpose we want (including refuting itself). A mechanism that can prove any claim is certainly not a reliable method of proof. And if it can prove a thing and its opposite, then it is certainly not an acceptable proof mechanism. This argument still does not explicitly present the flaw in this argument, but it does give us proof that there is such a flaw (just as in the liar paradox, where this is logically proved). If so, the conclusion that claim 3 is meaningless is no longer just an ad hoc assertion, but truly a proven claim.

But, as stated, I still have not explained what the problem is in such arguments. Ostensibly we have here a valid logical argument. The fact that it leads to absurd results only means that we should suspect it, and indeed that there really is some flaw in it. But we still have not found the flaw. The sting is that claim 2 here (and claim 3 in the original argument above) is in fact meaningless. We have not pointed that out directly, but indirectly we have proved it.

So that we are not left hanging in this way, I will add an aspect that will clarify the matter further, and I will return again to the question of the relation between the truth values of logical claims and the truth of their content.

On Logic, Truth Values, and Facts

Take, for example, claim A: "The chair beside me is made of wood." This is a factual claim. Now look at claim B: "Claim A is true." Ostensibly these are two claims with the same content, since both assert that the chair beside me is made of wood. But they are still not identical claims. The first makes a factual claim about the world, whereas the second says something about the first claim (namely, that it is true). The second is a claim about a claim and not about the world itself.

Notice that in the structures I presented in the two frames above, the first claims (claims 1 and 2 in the first structure, and claim 1 in the second) are factual claims, whereas the final claim (claim 3 in the first and claim 2 in the second) is a claim about claims (including itself, but that is not my focus right now). Notice too that the claim in the liar paradox is a claim about truth values and not about facts. The subject of the claim there is itself. But now I am focusing on another aspect: not self-reference (which, as I noted above, is not necessarily problematic), but the fact that this is a claim about claims and not about the world.

At first glance it seems that this is only a technical difference. On the substantive level there appears to be no difference between claims A and B, since both ultimately speak about a fact and assert something about it. But on a second look it becomes clear that the transition from facts (claim A) to logic (claim B) is not as innocent as it seems at first glance. It conceals pitfalls, and we must pay attention and be careful when making such transitions. In fact, we have already seen in the past two examples that illustrated the point, and so here I will only repeat them briefly. After that, we will return and see that herein lies the problem in arguments like the one with which I opened.

First Example: Logical Determinism

In column 301 I dealt with exactly this topic, and among other things I brought there as an example the argument known as "logical determinism." Aristotle discussed the claim "Tomorrow there will be a sea battle," and asked himself whether this claim is true or false. The truth value of a claim is determined by comparing its content to the state of affairs in the world itself. Therefore, in order to determine the truth value of that claim, we must wait until tomorrow and see whether there will be a sea battle (and then it will become clear that the claim is true) or not (and then it will become clear that the claim is false). But the need to wait for tomorrow is only because of our limitation. It has no connection to the truth value of the claim itself. Therefore, if the truth is that tomorrow there will indeed be a sea battle, then the claim is already true today, only I do not yet know it. The comparison between the content of the claim and the state of affairs in the world (which in practice can only be made tomorrow) reveals to us that this claim is true. And if so, there is no reason to refrain from saying that it is already true today. The conclusion is that in such a case the claim "Tomorrow there will be a sea battle" is already true today. In effect, we assumed here that the truth value of a claim does not depend on time but only on the comparison between its content and the state of affairs in the world. But if that is so, how can it be that tomorrow someone will decide not to hold the sea battle? After all, already today the claim is true, and if tomorrow the battle does not take place, it will turn out that the truth value of the claim is false rather than true. Here, then, is a logical proof in favor of determinism.

Notice that there is a double absurdity here: one who holds a libertarian view, that is, believes in free will, is troubled by the conclusion (the deterministic one). But even one who accepts determinism ought to be troubled by the fact that we proved a factual claim about the world by means of a logical trick (an argument consisting entirely of conceptual analysis, without any premises at all). This strongly recalls the argument we have been discussing here and the two difficulties I presented concerning it. I will not enter here into discussions and proposals about how to deal with this absurdity, and I will jump to my conclusion from there.

A logical trick cannot prove anything to us about reality in the world. Logic is supposed to be empty with respect to facts. In other words, logic deals with relations between claims and not with the content of claims. The content, that is, information about the world, must be learned from observation. Logical inference cannot teach us anything about the world. I argued that the fundamental flaw in that argument is the connection it creates between logic and the world. The fact that a certain claim has a truth value at a certain time does not necessarily say anything factual, and in particular does not mean that the information in question exists at that time. That is only a logical determination and not a fact. The truth value of claims is a definition of logicians, that is, it is a description of how we relate to claims and not a description of some fact. If I know now that tomorrow there will be a sea battle (say, from some reliable source), then indeed tomorrow a sea battle will occur. But if the truth value of the claim "Tomorrow there will be a sea battle" is already fixed today, that does not mean that what will happen tomorrow is already fixed as of today.

If tomorrow a sea battle takes place, then already today that claim is true; but if tomorrow no sea battle takes place, then the truth value of the claim is already false today. Perhaps it sounds strange to you that the future can determine the truth value of a claim in the past, but there is nothing strange about that. The truth value of a claim is not information and is not a fact (as noted, it is our definition). Therefore there is no impediment to the future determining a logical definition of the past. There is an impediment to the future determining information in the past, because causality works from past to future. But the relation between the fact and the truth value of the claim that describes it is not a causal relation (but a definitional one).

See there in that column for implications and arguments concerning free choice and more. For our purposes, the conceptual analysis we carried out of the structure of the three claims lies entirely in the logical sphere. To infer from it a conclusion about the world is a leap that requires justification. We will now see the same phenomenon in a different issue.

Second Example: The Logical Polygraph

In column 200 I dealt with the same topic from another angle. A logical schema was presented there by means of which one can extract the truth from any liar. Without entering into its intricate details (see there), I will describe it briefly here. I formulate a question that the person before me must answer as follows: if X is true, then he must answer "yes," and if not-X is true, then he must answer "no." What does "must" mean? If he answers differently, that drives us into a logical contradiction. But as I explained there, this still does not prevent him from moving his lips in a different way and answering "no" instead of "yes." The fact that I am driven into a logical contradiction will not really stop his lips. The conclusion there was that there is a principled problem in schemas that purport to extract a factual conclusion from logical-conceptual analysis. The world owes nothing to our logical and conceptual definitions, and to the relations among them.

Back to Our Case: This Is Only One Example of the Complex Relation between Logic and the World

In those two columns additional examples and implications of this principle were presented. For our purposes, this means that a conceptual analysis of a system of claims that we wrote on paper cannot yield factual conclusions about the world. And if in some strange way we succeeded in building a system from which a factual claim can be derived without premises, that only points to a flaw in the system we built or in our logic.

It can, however, be argued against this conclusion that we do derive factual conclusions from logical analysis. For example, in the cases mentioned above, regarding the existence of a round triangle, or regarding the fact that it is not true that something both exists and does not exist at the same time. In general, we tend to infer that if some claim contains a contradiction, it is not true, and therefore its expression in the world will not exist. This conclusion deals with facts in the world, even though its basis is a logical argument.

In columns 50 and 318 I addressed from another angle the relation between logic and mathematics and the world. In the examples discussed there, the problem was the formalization (the move from the facts to their formal presentation). The logical form does not represent in a reliable way the facts it purports to represent. Here, by contrast, the problem is the absence of factual content behind the formalization. These are collections of words that are not claims at all, but a façade.

Conclusion: The Different Faces of Ignorance

I will now return to the argument with which I opened. It begins with an argument that deals with claims and moves from there to claims about facts in the world. We have seen that this transition is hasty and liable to lead to failures, contradictions, and absurdities. When there is a logical argument that leads to a factual conclusion that seems problematic, and especially when the argument is ontological, we should suspect that some flaw has crept into it. And even if we have not managed to point it out, it still seems clear to me that it exists somewhere.

Interestingly, in the aforementioned video the speaker concludes by saying that common sense is not enough and one must study logic (the video is an advertisement or promo for a logic course he teaches). But from what we have seen here it emerges that in fact the lesson is the opposite: one who does not study logic is not troubled by these problems. Precisely if I am skilled in logic, I may get into trouble, because someone can present me with the argument he presented and entangle me. True, skill and understanding in logic can also extricate me from these problems, but a clever person does not get into problems from which a wise person manages to escape. We should not forget that the very engagement with these supposed fallacies and the entanglements we experienced here belongs to those who engage in logic. Ordinary folk are exempt from all this because they are not troubled by such arguments. It is intuitively clear to them that there is no problem, and therefore they do not need a solution either.

This raises the question whether I really became wiser after studying logic and, as a result, encountering this proof, and then using my logical knowledge to understand that its conclusion is false. How is my situation different from that of someone who has no interest in logic at all and knew from the outset that there was no problem? Does it not follow from this that logic is mainly a confusing tool? Does engaging this argument really give us a good reason to acquire logical skill?

This recalls the stale discussion about proofs for the existence of God as against simple faith. There too, the basic claim is: why get entangled with philosophy, which may raise various questions that a simple believer is not troubled by, even if in the end philosophical skill will help me prove that God nevertheless exists? Is it not preferable to avoid philosophy and thereby spare myself both the question and the answer? After all, from the outset I already know that God exists, so why get entangled in a field that raises difficulties and then solves them (even assuming that it really does solve them)?

In column 62 I discussed the ethos that glorifies ignorance in various fields. I explained there that even if ignorance has certain beneficial consequences, that does not turn it into something positive. And conversely, even if wisdom has problematic consequences, that only means that we must be careful in using it, but we should not infer from this that we ought to refrain from pursuing wisdom or from being wise. Elsewhere I explained that it is indeed true that excessive cleverness can lead to absurd conclusions, for intellectuals sometimes suffer from a lack of common sense, and in this matter ordinary folk can certainly contribute a perspective of common sense that intellectuals sometimes lack.

And for our purposes, the ignorant person is not troubled by the difficulty raised in the video, whereas the wise person is. An intellectual may draw the conclusion that indeed we do not exist, since he is committed to intellectual honesty and clings to logic more than to common sense. This is of course nonsense, and the ordinary layman, who sometimes is endowed with more common sense, can set him straight. But that only means that one must not be an intellectual cut off from common sense, but neither should one be a layman. The correct conclusion is that one should be an intellectual who is not cut off from common sense.

I have more than once written about the well-known saying (attributed to Rav Kook or to the Vizhnitz Rebbe) that it is preferable to fail in gratuitous love than in gratuitous hatred, whereas in my opinion it is preferable to fail in neither of them (see for example column 62). This is a very common pattern of argument, and of course a fallacious one. When there is a flaw in way A, that does not prove that way B is preferable. If it too has a flaw, one should aspire to way C, which contains neither type of flaw.

If you are aware of the difficulties and have solved them, you have not returned to the starting point. You have returned to your basic beliefs, but with greater depth and richer content. Your beliefs have become more meaningful and better grounded. A former friend of mine (before I became a heretic) used to say that if you resolved one of Rabbi Akiva Eiger's difficulties, you probably have an even number of mistakes. R. Akiva Eiger made one mistake when he failed to understand the Talmudic text. Apparently a certain insight escaped him. If you resolve his difficulty, it is unlikely that you are wiser than he was and that this insight is available to you. It is more likely that you made another mistake that cancels out the first one (or another three or five mistakes). In short, if one is going to err, it is better to make an even number of mistakes.

Is that really so? I think that if one arrives at the correct conclusion on the basis of a canceling-out of mistakes, it is not true that one really holds the correct conclusion. We think that we hold it, but that is only because we erred. It is well known that there are faulty arguments that lead to correct conclusions. Therefore, holding the correct conclusion does not justify false beliefs and faulty arguments that lead to it (see on this in column 52, on the difference between derush (homiletic exposition) and pilpul (sharp dialectic). Holding a correct conclusion by virtue of a faulty argument is derush. I wrote there that in my opinion pilpul is ten times better than it).