Trivial Rules as Negative Rules: On Psychology and Logic (Column 221)
With God's help
The first theorem in game theory was proved by the mathematician Ernst Zermelo in 1913. Zermelo's theorem states that in every game played in alternating turns between two players (a turn-based game), which is finite (ending after a finite number of moves), and of perfect information (each of the players knows the entire state, unlike Stratego, for example), and without chance (there are no random events, only decisions by the players, each on his turn, unlike backgammon, for example), there exists a strategy that guarantees victory to one of the sides: either player A has a winning strategy or player B does. Strange, no? It seems completely trivial. Obviously one of the only two possible results will obtain. What else could happen?!
There is a generalization of the theorem to games like chess or tic-tac-toe, in which the game may end without a decision, that is, in a draw (for example as a result of stalemate), which says that in such cases one of three possibilities may obtain: either White wins, or Black wins, or it is a draw. Again, this seems very strange: what else could happen in such a game?!
True, mathematicians devote many hours and thousands of pages to proving claims that look very simple, but here it already seems downright trivial. Prima facie, this looks like a tautology, not merely a claim that appears simple on its face. But on closer inspection you will see that the meaning of this theorem is somewhat different. The original theorem claims that there exists a strategy that guarantees one of the players victory. That is, it is not a claim about the result of this or that particular game (either White will win it or Black will), but a claim about the game as a whole. The claim is that either there exists a strategy that guarantees victory to White (White can force a win) or there exists a strategy that guarantees victory to Black. And in the generalization made for chess an additional third possibility is added: that there exists a strategy that allows each of the sides to force at least a draw.
Yet even this formulation looks trivial at first glance, since the theorem does not determine which possibility is the correct one, but only that one of them must obtain. Thus, for tic-tac-toe (which is a solved game, that is, one for which we know the strategy itself), the correct possibility is known to be the third. For Connect Four, the first possibility is correct. For chess, it is not known which of the possibilities is correct, and yet the theorem determines that one of them is correct there as well. So what, exactly, is the content of the theorem with respect to chess? Prima facie it is obvious that only one of these three possibilities can obtain, and the theorem does not even tell us which one. And even with respect to solved games, the fact that we found the correct possibility among the three is the result of a calculation. The mathematical theorem itself only determines that one of the possibilities must obtain (but does not give it to us). So what does it say at all? Is this not self-evident?
It is easier to understand the meaning of this theorem if we reverse the point of view and look at what it comes to rule out rather than at what it actually says. Let us try to think what we would say without the theorem. The logical inversion is that some game has no strategy of any of these three kinds. That is, the theorem comes to rule out the possibility that there is no strategy that allows one of the players to force any result whatsoever in the game. Or, in other words, every game proceeds however it proceeds and reaches whatever result it reaches, and even for a perfect player it is impossible to predict the result in advance. Sometimes Black will win, sometimes White, and sometimes there will be a draw. One cannot secure a definite result in advance. In other words, one might have thought that there is no strategy that determines the result beforehand, and the theorem determines that there is such a procedure (although it is not necessarily in our possession) such that, if one of the players uses it, he can force some result.
That is no longer trivial. Admittedly, there is still a strong intuition that this is true, and I assume most of us would bet that this is so even without knowing this theorem. But mathematicians do not make do with feelings. For them, what has not been proved is inadmissible (unknown). Incidentally, the proof of this theorem is fairly simple in terms of its methodology, but not technically simple (if I remember correctly, in the Open University course on game theory it takes almost an entire book to prove it).
Preliminary formulation of the conclusion
But my concern here is not Zermelo's theorem, nor game theory. What interests me is the general methodological conclusion that can be drawn from this: there are claims that seem trivial to us, but when we look at what they come to rule out, we better understand their meaning (why they are not really trivial). This is an important piece of advice in many discussions. When we examine some thesis, in many cases it is worthwhile to examine its negation, and through that to see exactly what the thesis itself says. Sometimes, when we examine the negation, we discover that it says nothing, and then the thesis turns out to be trivial. Sometimes we find that the negation is trivial, and then it turns out that the thesis itself says nothing. Quite a few philosophical discussions may turn out to be mere verbiage, if one examines the negation and sees that it has no real meaning. Sometimes the antithesis says exactly what the thesis itself says, and then it is only a word game. In the next column I hope to touch on these questions with respect to disputes between philosophical approaches.
Another example from the rules of Maimonides: ruling like the Jerusalem Talmud
Around every canonical text in Jewish law, a literature of rules has developed. There are the rules of the Talmud, the rules of the Mishnah, the rules of Maimonides, the rules of the Shulchan Arukh and the like. I regard most of them with great suspicion, because these are inventions that arose in the minds of commentators, not a rule that actually guided the author of the work himself. Regarding the rules of Maimonides, it is worth glancing at Rabbi Ezra Brand's article.
One of the rules of Maimonides seems to me especially suspect as lacking content. It is commonly thought that Maimonides generally ruled in accordance with the Jerusalem Talmud. And similarly with respect to the Sifra or the Sifrei against the Babylonian Talmud.[1] Thus, for example, the Ra'avad writes in Laws of Shema 3:6:
This rabbi's way is to rely on the Jerusalem Talmud.
When I was once asked about this rule, as can be seen here, I replied that in my opinion this is not the rule. The rule is that Maimonides sometimes (and not usually) rules in accordance with the Jerusalem Talmud.[2] That is probably the correct rule, but this formulation raises the same question that arose regarding Zermelo's theorem: what, exactly, does this rule say? That sometimes he rules in accordance with the Jerusalem Talmud and sometimes not. So in fact there is no such rule in Maimonides. In what sense can this statement be treated as a rule? What does this rule say?
And again, the right way to examine this is to check the negation: what does this rule come to rule out? On the logical level, it comes to rule out either the conception that Maimonides always rules like the Babylonian Talmud or the conception that he always rules like the Jerusalem Talmud. The claim is that neither is true. Maimonides is captive neither to the Babylonian Talmud nor to the Jerusalem Talmud.
As background, one must take into account the accepted approach that Jewish law always follows the Babylonian Talmud, whose main source is the Rif, who wrote (Eruvin 35b):
And we have seen that some of the great authorities hold like Ulla and rely on the Talmud of the people of the West, for we read there in tractate Yom Tov… But we do not hold this way, for since the sugya of our Talmud tends toward leniency, we are not concerned with what they prohibit in the Talmud of the people of the West, because we rely on our Talmud, for it is later, and they were more expert in the Talmud of the people of the West than we are; and had they not known that that statement of the people of the West was unreliable, they themselves would not have permitted it..
According to this, it seems that the purpose of this rule is to say that with Maimonides this is not the case. The rule does not come to say how he rules, but what he does not do. Maimonides does not obey the prevailing rule that Jewish law always follows the Babylonian Talmud. Because there is an accepted rule in the background, a rule that says Maimonides does not obey that rule definitely has meaning.
And indeed, this is what the Maharik writes in Shoresh 100:
And it is a well-known matter that Rabbenu Moshe regularly rules in accordance with the Jerusalem Talmud more than all the halakhic decisors known to us; and even in a place where our Talmud does not support the words of the Jerusalem Talmud, he will sometimes rule like them when our Talmud establishes a Mishnah or baraita by means of a forced interpretation, whereas the Jerusalem Talmud explains it according to its plain meaning—he adopts the approach of the Jerusalem Talmud..
It is quite clear that his intention is not to say that Maimonides' practice is to rule like the Jerusalem Talmud, but that his practice is not necessarily to rule like the Babylonian Talmud (as most other halakhic decisors do).
Another example from the Talmud: "half for yourselves"
The Talmud in Pesachim 68b cites a dispute among the tannaim regarding the character of a Jewish holiday:
For it was taught in a baraita: Rabbi Eliezer says: A person has on a festival only one of two options: either he eats and drinks, or he sits and studies. Rabbi Yehoshua says: Divide it—half for eating and drinking and half for the study hall. Rabbi Yochanan said: And both derived it from one verse. One verse says, "A solemn assembly to the Lord your God," and another verse says, "A solemn assembly shall be for you." Rabbi Eliezer holds: either all for the Lord or all for yourselves; and Rabbi Yehoshua holds: divide it—half for the Lord and half for yourselves.
Rabbi Yehoshua's opinion is understandable. In his view, on a Jewish holiday we must do both things: both rejoice and take pleasure, and also engage in spiritual matters. But what is Rabbi Eliezer saying? Prima facie he seems to be saying that it does not matter what we do: either we take pleasure or we engage in spiritual matters. But that is implausible, since Rabbi Yochanan brings a source from the Torah for both views. According to the explanation I suggested, R. Eliezer's position would not need a source, because he is not claiming anything. It is therefore clear that R. Eliezer is indeed demanding something of us, but from his words it is not clear what. It seems, at first glance, that one may do whatever one wants.
It seems to me that here too the way to examine this is to look at the logical negation. R. Eliezer comes to rule out Rabbi Yehoshua's opinion, who holds that we must divide the activities of that day. What R. Eliezer says is that it is forbidden to divide the day: either the entire day should be occupied with spiritual matters or the entire day should be occupied with pleasure and enjoyment. The verses came to teach us that the split is problematic. That is, his purpose is not to say that there are no halakhic demands and that we should do whatever we want, but that we must be consistent, and it is not right to divide the day between matter and spirit. This is also R. Eliezer's wording as Rabbi Yochanan explains it: 'either all for the Lord or all for yourselves' ('either entirely for God or entirely for yourselves'). It is clear from this that the intention is not that we should simply do whatever we want.[3]
Just for the sake of completeness, I will cite the continuation of the Talmud:
(Mnemonic: ayin-bet-mem.) Rabbi Elazar said: All agree with regard to Atzeret that we also require "for yourselves." What is the reason? It is the day on which the Torah was given. Rabbah said: All agree with regard to the Sabbath that we also require "for yourselves." What is the reason? "And you shall call the Sabbath a delight." Rav Yosef said: All agree with regard to Purim that we also require "for yourselves." What is the reason? "Days of feasting and joy" is written regarding it.
On Atzeret (Shavuot), on the Sabbath, and on Purim, even R. Eliezer agrees that we also require "for yourselves"—that is, enjoyment for yourselves. In other words, there there is an obligation of enjoyment. If on the other days R. Eliezer's reasoning is that it does not matter, that is understandable. On those three days it does matter: one must also take pleasure. But according to my suggestion, that R. Eliezer only requires that we not split the day, it follows that on Atzeret, on the Sabbath, and on Purim we should do only "for yourselves," because there is an obligation not to divide the day and there is an obligation that it also contain "for yourselves." The only way out is to make all of it "for yourselves," and this requires further investigation. Beyond that, one should note, conceptually: what is so bad, in R. Eliezer's eyes, about dividing the day between "for yourselves" and "for the Lord"? Why is it so important not to divide?[4]
Be that as it may, even if I am wrong, this is a good example of the principle we have seen here: a statement that sounds trivial can perhaps be interpreted in light of its opposite. One checks what it comes to rule out, and in that way understands it properly and sees that it has positive content.
On psychology and logic
The claim X and the claim that negates the negation of X ('it is not true that not-X') are logically equivalent, and so we would not expect there to be any difference between them. And yet we see that sometimes contemplating some claim through the negation of its negation can help us understand its content and meaning. On its face this is puzzling, since these are two claims that are logically equivalent. How does a difference arise between two identical claims?
It turns out that our psychology does not always cling to logic, and sometimes one formulation is easier for us to understand than another, even though they are logically equivalent to one another. I will now bring two examples of such phenomena.
A. The raven paradox
As is well known, Karl Popper already pointed out that one cannot prove a scientific theory. For example, the theory that all ravens are black cannot be proven (because even if we see several ravens and all of them are black, we can never be sure that we have seen them all. Perhaps there is another raven we have not seen that is not black). But it can be refuted: one non-black raven is enough to refute it. Therefore Popper determined that a scientific theory is a theory that can be refuted (as opposed to theories that can be neither proved nor refuted, which are not scientific).
Popper's method is that a scientific theory is subject only to refutation. But if so, it is difficult to understand why theories whose predictions were confirmed in experiment seem to us preferable to others that have not been refuted. Every theory that we use has many alternatives that we reject without their having been refuted. What is the justification for using one theory rather than the others, when its confirmation in experiment only says that it has not yet been refuted? Therefore other philosophers of science introduced the concept of 'confirmation,' which is softer than proof but still has weight. According to them, when some theory predicts an experimental result and carrying out the experiment shows that it was wrong, then the theory has been refuted. What happens if the predicted result is in fact obtained? According to Popper—nothing. The theory simply has not yet been refuted. But his opponents argue that the theory has been confirmed, that is, it has received further support (even though it has not, of course, been proven). In their view, the more experiments there are that verify the theory's predictions, the stronger it becomes. Thus, when we examine some raven and find that it is black, we have not proved the theory that all ravens are black, but we have confirmed it. If we find ten ravens and they are all black, it is confirmed even more. In their view, a confirmed theory is preferable to theories that have merely not been refuted, and the more cases in which it has been confirmed, the more preferable it is. This is the confirmation thesis, which was set against Popper's conception.
Against this thesis, the philosopher Carl Hempel (in the 1940s) raised the raven paradox. He made the following claim. Consider the theory: all ravens are black. This theory is confirmed by observing a raven and determining that it is black. A formulation logically equivalent to this theory is: everything that is not black is not a raven (check and see that this is equivalent). From the confirmation thesis it follows that every non-black object that we examine and find not to be a raven confirms this theory. And from this it follows that observing a pink table, which confirms the thesis that everything non-black is not a raven, necessarily also confirms the thesis that all ravens are black (to which it is equivalent). According to Hempel, it is absurd to think that finding a pink table confirms the thesis that all ravens are black, and therefore it is clear that the claim about the confirmation of a scientific theory is senseless.
I assume that the overwhelming majority of readers will agree that finding a black raven does indeed confirm the theory that all ravens are black. Our science is built on that (the thesis that all of science is nothing but a collection of theories that have not yet been refuted is absurd). On the other hand, all of them will also agree that finding a pink table does not confirm this theory. But it is not clear how it can be that finding a pink table confirms the theory that everything non-black is not a raven, but does not confirm the theory that every raven is black. After all, these are two formulations that are logically equivalent.
Surprisingly, the solution is that finding a pink table really does confirm both formulations of the theory. Logical equivalence is cast in concrete, and there is no way to bypass it or break through it. The problem here is not in logic. To understand this, we must look at the negative formulation: everything that is not black is not a raven. What experiment must be carried out in order to confirm this theory? We must look at an object that is not black, and examine whether it is a raven or not. In other words, it is not enough to look at a table and see that it is pink. That is not an adequate confirmation of the theory. We must look at an object that is not black (before we know what it is), and then check whether it is a raven. If it is not a raven, then we have indeed confirmed the theory that everything non-black is not a raven, and thereby also the theory that all ravens are black.
What is confusing here is that there are many more objects, and many more kinds of objects, that are not black than there are ravens. If there are a million ravens in the world, and they all belong to the same type of object, then the collection of non-black objects presents a far greater variety of objects from many kinds. Therefore the theory that says something about them is much harder to confirm. Every observation we make (we examine a pink object and discover that it is a table and not a raven) can be a particular and exceptional case: perhaps only tables are not black, whereas other objects that are not ravens are black? The generalization that would be made here on the basis of the experiment is far less well-founded. But generalization is not a logical-deductive process, and there is no reason to assume that generalizing from one black raven to all ravens will have the same force and the same reliability as generalizing from a pink table to all non-black objects.
This is a fine example of the fact that two claims that are logically equivalent are not perceived by us in the same way. It is easier for us to handle one than the other. The difference between the claims is not on the logical plane (for they are equivalent to one another), but on the psychological and scientific plane. The scientific difference lies in the nature of the groups involved. But beyond that there is also a psychological difference, for it is easier for us to understand that a black raven confirms the theory that all ravens are black than to understand that finding a pink table also confirms that same theory (though to a far lesser degree). The positive formulation is psychologically more accessible to us than the negative formulation, despite their logical equivalence.
B. 'Theological' proofs
In the fourth notebook I distinguished between two kinds of arguments: a 'philosophical' argument—which proceeds from the premises to the conclusion—and a 'theological' argument—which proceeds from the conclusion to the premises. I explained there that both kinds of argument are logically valid, and they are based on an equivalence between two kinds of propositions. The proposition: A -> B (the arrow expresses implication), is equivalent to the proposition: B -> A (the underline indicates negation).[5] I brought there two such pairs of arguments (the argument from epistemology in the second part and the argument from morality in the third part), and to illustrate our point we shall now look at the second pair.
In the third part of the notebook I dealt with the moral proof for the existence of God, and I argued there that without God there is no validity to morality. A valid moral norm receives its validity from some source (which cannot be human beings themselves); otherwise it is merely a form of behavior and not a binding norm. And from this it follows that a valid moral norm cannot exist in a world in which there is no source that can give it validity (which I called God). This can be formalized as follows:
A – God exists.
B – There is valid (binding) morality.
The basic claim is: A -> B (if there is no God, there is no morality). A claim logically equivalent to this is the following: B -> A (if there is morality, there is God). The second formulation is much less intuitive, and already here there is a hint that our psychology does not necessarily overlap with logic. Sometimes, of two equivalent formulations, one is more accessible to us than the other.
The claim usually made against atheists is that there is no validity to their morality. This is a 'philosophical' argument. I suggested there a parallel 'theological' argument: if you believe in morality, you necessarily believe in God (even if you are not aware of it, there is an implicit belief within you). I showed there that these are two different kinds of claims. From the first it follows that the atheist must give up his morality (or his atheism), while from the second it follows that the person who believes in morality must give up his atheism (or his morality). See there for the implications and differences between the arguments, and the different forms of evasion from them. Bottom line, I will not go into all that here, but one should notice that these are two arguments that are logically equivalent, and yet their implications for us are different. There are cases in which a person will be convinced by one and not by the other, or vice versa. Again, this is a difference in psychological accessibility, not in logic.
Put differently, one can say that the matter depends on the question of what is perceived by the moral atheist as the stronger anchor. Both arguments point out that those two things (morality and atheism) do not fit together, that is, that he must give up one of them. The 'philosophical' argument addresses the atheist for whom atheism is stronger and draws his attention to the fact that he must give up his morality. The 'theological' argument addresses the atheist whose morality is firmly anchored and tells him that he must give up his atheism. That is, the difference lies in our a priori assumptions as human beings, not in the logic of the arguments themselves.
[1] In a note in Rabbi Brand's article he writes:
Maimonides sometimes even rules in accordance with the Jerusalem Talmud against the Babylonian Talmud, as the Gra noted several times in his commentary to the Shulchan Arukh; see a list of these places in Mishneh Torah, Frankel edition, Sefer Ahavah, Jerusalem 2007, p. 340. See there for further sources on this, and also in Shapiro, Studies, p. 2, note 4. However, the commentators take it as obvious that Maimonides would not rule in accordance with any other book—apart from the Jerusalem Talmud—against the Babylonian Talmud. See Yad Malakhi, Rules of the Decisors, Rules of Maimonides, sec. 9, that Maimonides certainly does not rule like the Tosefta or Sifra or Sifrei against the Babylonian Talmud; at most, the Tosefta or midrash can serve as a source for Maimonides' words or reveal another plain meaning in the words of the Talmud, but not explain the omission of a law that is explicit in the Babylonian Talmud (however, see there in Yad Malakhi, who cites Knesset HaGedolah to the effect that it is possible that Maimonides would rule like the Sifra or Sifrei against the Babylonian Talmud).
[2] And so too in Maharik, Shoresh 100, and in other sources brought here. Some wanted to say that he ruled like the Jerusalem Talmud in places where it contains a clear ruling (such is the opinion of several scholars), and in the aforementioned article there is a different explanation. But without my having checked the matter to the end, it seems to me at first glance that neither is persuasive.
[3] It would have been possible to explain this wording, somewhat forcedly, as coming to add two more possibilities: either half for yourselves and for the Lord, or all for yourselves or all for the Lord. That is, the other two possibilities would also be legitimate. But this does not seem to follow from the flow of the Talmudic passage, because then we would have expected Rabbi Yehoshua's opinion to be brought first, and then R. Eliezer to add his two possibilities. Beyond that, if this were R. Eliezer's intention, we would not have needed to bring him a source from the verses.
[4] It is interesting that the two verses brought in the Talmud as a source, one deals with the seventh day of Passover ("a solemn assembly to the Lord your God") and the other with Shemini Atzeret ("a solemn assembly shall be for you"). Prima facie, one might have learned from them that these two days have a different character, rather than seeing them as two poles that exist in all the festivals of the year.
[5] Incidentally, this equivalence is the same equivalence we encountered in the raven paradox. There the equivalence was between the claim 'every A is B' and the claim 'every B is A.' Notice that when we say every A is B, what we have really said is that if something is A, then it is necessarily B (A -> B), and thus we arrive at the second equivalence presented here.
Discussion
And when I said “generalizes,” I didn’t mean to the other games, but rather the search for special strategies in different kinds of games (not necessarily winning ones, like fixed points for example. Maybe that is what the rabbi meant, since they are like a kind of draw).
In my view, an inductive proof gives excellent intuitions as to why the theorem is true.
It’s ridiculous to think that finding a pink table confirms the claim that all ravens are black only if we know it is a table after we know that it is not black. That’s like saying we first need to see the raven and then check whether it is black. The more black ravens I find and not a single white raven, the more I confirm the theory that all ravens are black; and the more things I find in various colors that are not ravens, the more I confirm the inverse of the theory – that what is not black is not a raven. But note that whereas there is a limited number of ravens and of black things, and therefore finding a black raven confirms the theory, there is a huge number of non-ravens and of non-black things, so the degree of confirmation is dozens of times smaller. (As you said in the following paragraph.)
And there is a mistake in the post – “Every observation we make (we examine a pink object and discover that it is a table and not a raven) could be a particular and exceptional case: perhaps only tables are non-black, but other objects that are not ravens are black?” If I find a black table, that would not refute the theory for me. The theory says that all ravens are black and that everything non-black is not a raven. It does not say that only ravens are black and that everything black is a raven. Therefore there are 4 possible findings:
1. A black raven – confirms the theory
2. Something non-black and not a raven – confirms it to a lesser degree
3. A raven that is not black – refutes the theory
4. Something black that is not a raven – has nothing whatsoever to do with the theory.
In addition, we haven’t touched at all on the different forms of inversions.
The statement – all ravens are black – can be inverted in several ways:
1. All ravens are black
2. Everything that is not a raven is not black
3. Only black things are ravens
4. Only non-black things are non-ravens
All four are logically equivalent of course, but if you had used all of them it would have been easier for me to understand the article.
And I’ll end with a question that has long bothered me, and to which I have found no answer to this day: why is a raven like a writing desk?
Maybe the truth is ridiculous, but that does not mean it isn’t true. It doesn’t matter what you actually do, so long as the outcome is not clear to you in advance. If you choose in advance an object that is not a raven and find that it is not black, that is not like choosing an object that is not black and finding that it is not a raven. If you choose an object that is not a raven and know in advance that it is not black, that has no weight whatsoever (not even epsilon).
As for the “mistake”: who was talking about refutation? I did not understand your bizarre claim.
As for the different forms of inversion, you are of course welcome to write a follow-up column and discuss them. What does that have to do with what I said?
By the way, this relation between a positive proof that all ravens are black and a negative proof that all non-black things are non-ravens is a bottom-up vs. top-down relation, or from the inside out vs. from the outside in. There are many such definitions of mathematical structures (mainly in algebra and topology) that are defined on the one hand as combinations (via algebraic operations) of specific objects from the set, or as the smallest set contained in all the sets that contain those elements and satisfy certain axioms (the field axioms), and which is always their intersection (like a subfield generated by certain elements in a field, called generators, which is the collection of all sums and products of those elements and their inverses, and on the other hand the smallest subfield containing those generators; or a subalgebra of sets (an algebra in measure theory) generated by the collection of complements and finite unions of the generators, or as the smallest subalgebra satisfying the algebra axioms and containing the generators). In any case, they always give both definitions, which are logically equivalent, in order to provide a full understanding of the structure.
Interestingly, usually one of them is the one usually presented as the definition and intuitively fits that role, while the other is presented as a characterization (a logically equivalent property). Here too there is a scientific hierarchy between positive confirmation and negative confirmation, which is weaker but still necessary in order to provide stronger and fuller confirmation. It seems to me that here too the theological proof and the philosophical proof are a kind of complementary proof of the same essence (the unified entity of God and morality).
Correction to the end of the comment: constitute a kind of complementary proof of that same essence.
Right. On a further reading I see that you said everything I said in the first paragraph anyway.
If you weren’t talking about refutation, I’d be glad to know what you were talking about – that was the only context I could find connecting claims about the set of non-black objects and black tables. “Perhaps only tables are non-black but other objects that are not ravens are black?” I didn’t understand this sentence and what it connects to:
There it spoke about a claim regarding all non-black objects, and you didn’t specify what that claim was. From the context I assumed the claim was that they are not ravens. (It may be that this is my mistake.)
After that you said we examine a non-black object and discover that it is not a raven.
And then you said that this could be a particular and exceptional case – perhaps there are black things that are not ravens. That is the claim whose place in the paragraph I didn’t understand, and I’d be happy for an explanation.
I said there are different forms; I don’t see anything to add beyond their existence – this column explains well the principles that apply to all the inversions.
Isn’t all this summed up in the clarificatory question: “What practical difference does it make?” For if there is no practical difference between one claim and another, then nothing practical has been said.
I didn’t understand.
I said that because there are quite a few kinds of non-black things, if you found a pink table, it is hard to infer anything from that. Because it may be that tables have some special property of being non-black, and that says nothing about other kinds of non-black things that could be ravens. That is not a refutation but a weakening.
To Rabbi Michi,
What do you say about the following argument:
In the statement “All ravens are black” – what do you mean when you say “raven”? There are only 3 options:
(1) Something that is necessarily not black.
(2) Something that is not necessarily black.
(3) Something that is necessarily black.
If you mean (1), then you are stating a contradictory proposition.
If you mean (2), then you are stating a contradictory proposition too.
If you mean (3), then you are stating a tautological proposition.
The same move can be made with any scientific law.
(Perhaps one could answer, “A raven that until now we thought was (2) is really (3)!” But why would we say that?)
You describe the gap between confirming the thesis and confirming its equivalent as a quantitative gap, without a difference in essence.
But for every false and refuted theory one can confirm its equivalent infinitely many times – try, for example, “All fairies are black.”
If the ability to confirm the claim is 0, but the ability to confirm its equivalent is infinite (lots of non-black things that are not fairies), that points to a problem in your statement in the article that the more a theory has been confirmed, the better it is, (unless you make an essential distinction between confirmation of the claim itself and confirmation of its equivalent).
I would say the argument seems very strange to me. A raven has a biological definition, and its color is not part of it. The claim “All ravens are black” means that whoever satisfies that biological definition is black in color. Neither a contradiction nor a tautology.
I very much disagree. Of course false claims can be confirmed. That is why it is confirmation and not proof. But what does this prove? In analytic philosophy they have already discussed the sentence “The present king of France is bald.” If you look for him in the set of bald people, you won’t find him. But you also won’t find him in the set of hairy people. Simply because France has no present king. Fairies do not exist either, and therefore that example is not relevant to the discussion.
A raven has a biological definition? I haven’t heard of it, and I doubt it exists. Which part exactly of the DNA would have to be missing for it not to be a raven? In my humble opinion, “raven” is a word that comes to describe something in reality by pointing to phenomena that we perceived with our minds as similar (no two ravens are identical); as far as I know, Wittgenstein held this too.
I was not relying on the fact that false claims can be confirmed (and therefore confirmation is worthless), but rather on the fact that there is a class of false claims (things that do not exist, for example) that, although they cannot be confirmed at all, their equivalent can very easily be confirmed,
This is supposed to confirm (not prove) the claim that there is an essential difference between confirming the claim and confirming its equivalent…
There is a simple solution to your not having heard of it. Go study. I do not see why ignorance is an objection.
You certainly did say that false claims can be confirmed, by confirming their logical equivalent. And to that I answered you that bringing an example of nonexistent creatures disrupts the whole scientific procedure. It proves nothing. It neither confirms nor proves anything about confirmation of scientific claims, other than that confirmation is not proof (and therefore sometimes it doesn’t work). I already wrote that to you too.
Another formulation. You claim that the statement all fairies are black is logically equivalent to the statement everything non-black is not a fairy. But how can there be two logically equivalent claims, one false and the other true? Equivalent claims must have the same truth value. That is the definition of logical equivalence. When the statement concerns nonexistent creatures, the logical equivalence breaks down (as in the example of the present king of France, where the law of excluded middle breaks down).
Of course, you can define the statement “All fairies are black” vacuously as a hypothetical statement: if there is a fairy, it is black. That is indeed a true statement, and confirmation of its logical equivalent confirms it as well.
But all these are unnecessary pilpul.
I will go study. But even if I find the most up-to-date definition of a raven, it won’t matter, because that definition was created long after the word itself was created. Did people in the biblical period think of the biological definition when they said “raven”?! No. And yet you would say that even if the sentence “All ravens are black” was said then, it still means what it means today after there is a definition. (By the way, in some dictionaries part of the definition of “raven” is “black,” so in this case the example really is very poor.)
If the logical equivalence breaks down, then indeed what I argued is not relevant.
In any case, I enjoyed the article. Thank you.
After they discovered that ravens are black, they put it into the encyclopedia entry. By your method, that whole entry is unnecessary, because everything said there about ravens is a definition and not a claim. You have erased the difference between an encyclopedia and a dictionary.
It seems to me this pilpul has pretty much run its course.
With God’s help, 15 Sivan 5779
Ketubot 49a distinguishes among ravens between white and black ones. According to Rashi, this refers to different stages of the raven chick, which is white at first and then blackens. According to Tosafot (based on a passage in Hullin), these are two species. There is a ‘black raven,’ and there is a ‘valley raven,’ which is white (according to the Ran, not completely white but speckled). I discussed this at length and cited sources in my response to Tsachi Cohen’s article “Not All Ravens Are Black,” on the “Shabbat Supplement – Makor Rishon” website.
Regards, Shatz
See also Dr. Moshe Raanan’s article, “The Raven Wants Its Young,” on the “Portal of the Daily Daf,” Ketubot 49b.
The description of white chicks that blacken could fit the starling (the raven’s neighbor and relative), whose feathers are black during the breeding season and fade at other times. Rejection of chicks exists in the common starling for chicks from the second and third laying, whom the males refuse to feed if there are chicks from the first laying. See Wikipedia, entry “Common Starling.”
Regards, Shatz
It was not for nothing that the starling (the common one) went to the raven (the black one), but because it is of its color.
I saw several puzzling comments here that suggest a lack of understanding of concepts in epistemology (and in my humble opinion Wittgenstein failed in this as well), and above all a failure to distinguish between a percept and a concept.
Sense data are specific and particular (there is no generality in sense perception), therefore one winged creature is not another creature even if it resembles it like two drops of water; from the standpoint of the senses these are two entirely separate things even if they are similar. A concept, by contrast, is a general idea, a creation of our intellect (we invented it), and we determine what the content of that idea will be according to what we decide (therefore Wittgenstein’s words, that a concept is a collection of sensory phenomena received by us, are very puzzling indeed. A concept is an idea whose content must be defined; after we define it, we can examine which sensory phenomena fit the idea that we ourselves created).
Therefore, if we decide that the definition of the concept raven includes the color black, then so it will be, and then any creature that is entirely identical to our black ravens will not be included in the concept raven. However, if we decide that color is not relevant in the concept we created, but rather other properties are, then so it will be.
In any case, there are two important points here.
1 A concept is not a percept, because a concept is an idea, and it needs to be properly defined.
2. The definition of a concept is arbitrary; nobody can argue with me about what the correct definition of a particular concept is.
I don’t know what you saw here or what your remarks are referring to.
I agree with most of what you say here. I do not agree that the concept is a creation of our spirit, and for the same reason I do not agree that it has nothing at all to do with the percept. The percept is the basis from which our cognition distills the concept, and afterward we go back and find the percepts that fit the concept that was formed. Moreover, in my view the concepts exist as ideas, and we “observe” them (with the “eyes of the intellect”; see Maimonides, beginning of the Guide for the Perplexed. This is Husserl’s eidetic intuition). The percepts help us discern the concepts.
Therefore:
1. Entirely correct.
2. Not correct. A definition is arbitrary, but a concept is not. The definition does not constitute the concept; it tries to hit upon it. If there is no concept in reality (a correlate), then the definition is arbitrary and then one cannot argue about it. But there are definitions that aim toward reality, and about those one certainly can argue. I have already elaborated on this in several places.
But I do not see what all this has to do with what was said in my column here.
That is what seemed to me to underlie one or two disputes above, and therefore I wrote what I wrote.
2 I did not understand your distinction. Even concepts that refer to reality are products of the intellect. It is indeed true that the decision about which properties will enter the definition of the concept is influenced by sensory data, but it is not determined by them (as I understand it, determining the properties that create the definition of the concept is an entirely semantic matter whose main point is the usefulness that the definition of the concept gives us).
Perhaps I’ll formulate the question from another direction: according to you, a concept is an objective reality that exists outside us, and we try to hit upon it with our minds. Whereas I claim that even a concept that refers to external reality is only our own subjective idea, which comes to describe different collections of sensory data that were fixed at the basis of the definition of the concept in question. I do not see how one can claim that a certain concept is something outside us; I would be glad for a further explanation or a reference.
I still have not understood what your remarks are aimed at. Why is this discussion connected to my column?
As for the matter itself, my claim is that the definitions of concepts are not completely subjective. I am an essentialist (and you are a conventionalist). An illustration of a strong argument against this conventionalist thesis is found in Borges’s wonderful story, “Tlön, Uqbar, Orbis Tertius” (in the collection Fictions). In brief, the claim is that if indeed the definition of a concept is a convention, then to the same extent one could define a concept that includes the pitch of a bird’s cry at a distance together with the temperature level in Oceania and the amount of rainwater in the Mediterranean Sea, and create from that aggregate a concept. For a concept is nothing but a collection of characteristics put together as we wish (convention). That is unreasonable.
When we define, for example, democracy as a regime in which there are civil liberties, elections and voting rights, civil rights, separation of powers, and the like, this is not only a matter of convenience. There is a real connection among these characteristics, and there is something (an idea, an abstract object) that has these properties. That is what makes them belong to one concept. Therefore here it makes sense to create a concept from this collection, whereas in the previous example (the bird’s cry…) it does not.
Beyond that, it is simply a fact that people argue about definitions (such as who is a Jew and what is moral). According to your view, these arguments have no meaning. So-and-so will define the concept this way and someone else that way, and they will part in peace. From this it follows that both sides agree on one thing: that there is an idea whose definition this is. The argument is about the definition of that idea. Therefore there is a real dispute here.
You can of course claim that in this both sides are mistaken, but I am offering an explanation of why that is not so. We all have the intuition that there is an idea whose definition is what the dispute revolves around, and therefore it is not a fictitious dispute. And if we have such an intuition, then apparently that is how things are until proven otherwise. I do not disdain people’s intuitions (as opposed to arguments, where sometimes there really are stupid things).
When you ask me for a reason, I will answer you that there is also no reason for the trust you place in what you see. You simply see, and that is that. And one who does not see is mistaken (blind). Here too I would answer the same. One who does not see the idea is simply mistaken. I do not need to explain the trust I place in my “seeing.” By the way, in my opinion the mistake is usually that people do not understand that what they sense is not a subjective matter but the product of seeing with the “eyes of the intellect.” Because the concept is unfamiliar to them, they think there is no such thing, and therefore in their opinion it is merely a subjective matter. That itself is the mistake of conventionalism.
Thanks for the explanation!
Still, I didn’t understand, and I’d be glad for further clarification.
This is what was said in the article: “Perhaps only tables are non-black but other objects that are not ravens are black?” (that is, perhaps there are objects that are not ravens but they are black – which has nothing to do with the theory ‘all ravens are black’)
And this is what was said in the response to my question about the meaning of the paragraph: because it may be that tables have some special property of being non-black, and that says nothing about other kinds of non-black things that could be ravens. (that is, perhaps there are non-black things that are ravens. Which means the table alone does not confirm the theory well.)
If these are talking about the same thing, I can’t understand how there isn’t a mistake here – the paragraphs are making completely different claims, and to the best of my understanding the second is the explanation of the first and not of some other paragraph.
(After all, you yourself said that if we want to confirm the theory that all ravens are black, we should examine colored objects to see whether they are ravens, or ravens to see whether they are black. Examining black objects to see whether they are ravens, as you suggested here, or examining non-ravens to see whether they are black, is not effective for confirming the theory, and is worthless for refuting it.)
There are no disputes about definitions. The disputes are about public opinion. After it has been implanted (through brainwashing) into public opinion that democracy is a good thing, all kinds of thieves come and try to inject their own content into it.
The dispute is about public opinion, not about the definition.
There certainly are disputes about definitions. Who is a Jew, what is democracy, and so on. You can only insist on your a priori position (conventionalism) that these are pseudo-disputes, but as I wrote, the burden of proof is on you.
I can’t deal with such gaps in the discussion. My principal claim still stands: when there is a variety of species under a genus (such as a variety of things of several species under the genus of what is not a raven, or of everything that is not black), then confirmation regarding one species (finding a pink table, that is, a non-black object that is also not a raven) constitutes weak confirmation regarding the whole genus (that is, the claim that everything non-black is not a raven), and therefore also of the equivalent positive thesis that all ravens are black.
Okay. I’ll bring it from the beginning briefly:
It seems to me there is a simple mistake in this paragraph: “What is confusing here is that there are many more objects and many more kinds of objects that are not black than the number of ravens. If there are a million ravens in the world, and they all belong to the same kind of object, then the collection of non-black objects presents a variety of many more objects from many kinds. Therefore the theory that says something about them is much harder to confirm. Every observation we make (we examine a pink object and discover that it is a table and not a raven) can be a particular and exceptional case: perhaps only tables are non-black but other objects that are not ravens are black?”
The mistake is that a claim about the color of objects that are not ravens is not relevant to the theory; that is, the last line should be corrected to: “…but other objects that are not black are ravens?”
Indeed.
With God’s help, 21 Sivan 5779
To R. M. A. – greetings,
What you noted, that the starling resembles the raven in its black color – according to what I cited from the Wikipedia entry “Common Starling,” the starling’s blackness is specifically during the breeding season, while outside that season its color is dingy white.
Perhaps this borderline quality of the starling in resembling the raven is what led to the dispute among the Rishonim whether to define it only as shakhen (“dwelling with/neighboring”) the raven, or also as nidmeh (“resembling”).
In Hullin 65a a baraita is brought: “Others say: one that dwells with impure birds is impure; one that dwells with pure birds is pure,” and the Gemara identifies “Others” as Rabbi Eliezer, who says: “It was not for nothing that the starling went to the raven, because it is of its kind.” And they rejected this: “You may even say the Rabbis, for we require both dwelling with and resembling.” The Ran concludes that the halakhah is not in accordance with Rabbi Eliezer that the starling is impure, for we rule like the Rabbis, who require specifically both “dwells with” and “resembles.”
However, Maimonides brings in Hilchot Ma’akhalot Assurot 1:14 Rabbi Eliezer’s exposition: “for with the raven it says ‘according to its kind’ – to include the starling,” and in halakhah 19 he seems to adopt the Rabbis: “And anything that dwells with the impure and resembles them is impure.”
The Kesef Mishneh explained that according to Maimonides, even the Rabbis agree with Rabbi Eliezer that the starling is forbidden, since the starling also “resembles,” and their dispute is only about the category of prohibition: according to Rabbi Eliezer it would be forbidden even without resemblance.
R. Yosef Kapach cites (on p. 37), in the name of “Tziyunim of the Maharan,” the version of Halakhot Gedolot in Hullin 65, according to which “Others” spoke explicitly about the starling, and to this the Gemara said: “You may even say the Rabbis, when it both dwells with and resembles it.” According to this version, it is proven that the starling too is considered as “dwelling with and resembling.”
And R. Yosef Kapach concludes (on p. 39): “And it is close to accept what the Tziyunim of the Maharan wrote, for when there is a text that requires no explanation, it is certainly preferable according to Maimonides’ method.”
Regards, Shatz
R. Yosef Kapach’s rule, that in explaining Maimonides one should prefer the straightforward text that does not require a forced explanation, also accords with Maimonides’ way of ruling like the Jerusalem Talmud where its explanation is simpler (as you cited in the name of the authors of the rules).
On the importance of the Jerusalem Talmud, Maimonides says in his introduction to the Mishneh Torah: “And from the two Talmuds, and from the Tosefta, and from Sifra and Sifrei and from the Toseftot – from all of them will be clarified what is forbidden and permitted, impure and pure, liable and exempt, valid and invalid.” And although Maimonides mentioned that the Babylonian Talmud is the “end of instruction” because it was compiled last, he nevertheless attributed great importance to the Jerusalem Talmud and even composed on it a digest of the halakhot similar to what the Rif did on the Babylonian Talmud (and from Maimonides’ halakhot on the Jerusalem Talmud, the tractates Berakhot and Ketubot have survived, published by R. Shaul Lieberman).
Paragraph 7, line 1
…“And it is close to accept what…”
Paragraph 8, line 2
… where its explanation is simpler…
The Raavad wrote: This rabbi’s way is usually to rely on the Jerusalem Talmud, and here he did not rely on it. Meaning: generally Maimonides’ way is to rely on the Jerusalem Talmud as a source for halakhic rulings, and here he did not rely on it, etc.
That is, here he did not rule like the Jerusalem Talmud, which forbids reciting opposite the urine of a donkey coming from the road, as well as the red droppings of chickens. But the Beit Yosef wrote that apparently Maimonides did not have this in his text (as in our text), but rather “donkey dung,” and he included donkey dung and red chicken droppings in the category of any excrement with a foul smell. It follows that here too he relied on the Jerusalem Talmud.
Note: at the Open University, the proof of Zermelo’s theorem (for chess) takes barely 2 pages. And if I remember correctly, it’s an inductive proof (which is a bland proof that provides no insight into why the theorem is true). But it is true that the book generalizes this theorem again and again (von Neumann’s minimax theorem, Nash’s fixed-point theorem) all the way to the end of the book.