The claim is that even though we have a great deal of trust in a certain sense, it won’t really be effective for a rare claim. When the rarity level of the observed thing is on the same order of magnitude as the chance of error in the sense, then the chance that the sense is correct is almost meaningless.
You can think of the same idea with a medical test that is supposedly quite reliable, with a 99% success rate, trying to detect a disease that occurs in one in a million. The chance that the person found to be sick really has that disease is negligible.
The same thing happens with the sense of sight. Suppose your confidence in it is 99.9%, but you observe a rare object (for example, a snake on the road); the chance that it really exists in front of you is completely negligible.

I wrote an article about this phenomenon:

כֶּשֶל היציגות בהלכה

But in your example it isn’t relevant, for several reasons. First, the error in vision is small, so if you see a pen, at most it might perhaps be a pencil, but not a table (and even that with a very small probability). Beyond that, the appearance of objects before you is not random.

I didn’t understand whether you are claiming that our trust in our eyes is close to the number of objects in the world (with 99.999……. digits), which is completely unreasonable to claim…
Or perhaps you meant that trust in the eyes is only one-directional, but that too is not exactly reasonable to claim.

As for the second claim, I think it is not essential, and if you like, think of a captive person whose eyes were tied shut and who was thrown by parachute from a plane; from his perspective, whatever he sees afterward can be regarded as a random situation.

With all due respect, this pilpul is bizarre. The error in vision is dramatically smaller than the number of alternatives that could appear before us. When I look at a pen before my eyes, there is no chance that in reality what is there is a plane flying in the skies of Australia.
I think we’ve exhausted this.

And therefore I meant the whole sense of sight in its entirety, in which case one cannot claim that the number of alternatives is small.
It is worth mentioning that you devoted a lot of time and space to a similar pilpul in dozens of pages of the fourth booklet… and in your hefty book.

In connection with this question, I wanted to ask about translating a doubt in judgment into a probabilistic doubt, if that is possible. For example, in the legal world there is a term called “beyond a reasonable doubt.” Some people would prefer a more mathematical definition like “beyond a doubt of X percent.” Recently I thought that this translation is impossible. Because doubts in judgment are by definition not doubts of the computational type that a computer can handle (as you said in the responsum “Choosing to Believe” a few days ago). Therefore, by definition, it is impossible to assign them a probabilistic value.

Let me give another example to sharpen what I mean. A person can ask himself: at what level of likelihood do I believe in God? Maybe 50%, maybe 80%, maybe 99%. I claim that the plausibility level of faith cannot be translated into percentages; rather, we have a completely different kind of scale here, a non-numerical scale. The markings on the scale have verbal rather than numerical values, such as “likely,” “unlikely,” “evenly balanced doubt,” “patently unlikely,” and so on.

This reminds me of what you once wrote about the IQ scale, where you argued that it is an ordinal scale and not a counting scale, or a qualitative rather than quantitative scale, unlike the temperature scale. The same applies to scales of plausibility.

Maybe this helps explain the Jewish law that in a fixed case, it is treated as half-and-half. Many ask how it can be that a doubt whose probability is, say, 10% should be treated like a 50% doubt. But according to what I said above—that there is no simple translation between the verbal scale of plausibility and the numerical scale of probabilities—it is possible that even a doubt whose probability is only 10% falls under the definition of an evenly balanced doubt (which is what “it is treated as half-and-half” means).

There is another implication of my claim regarding the laws of double doubt. Some argue that a double doubt amounts to multiplying probabilistic doubts. So if I have a double doubt made up of two evenly balanced doubts, then ostensibly the double doubt should have a probability of 25% (0.5*0.5). But according to what I said above, one cannot perform a simple mathematical multiplication on these doubts, because they are not “probabilistic” doubts but “plausibility” doubts. Therefore, from a halakhic standpoint it is possible that a Torah-level doubt whose source is a fixed case, and whose law is that it counts as an evenly balanced doubt (half-and-half), will be ruled stringently even though its probability is ostensibly 10%, whereas a double doubt in a Torah-level case whose probability is ostensibly 25% will be ruled leniently. The reason is that mathematical language is not the correct language for dealing with doubts, and that is what causes the confusion on this topic.

I also recall that you once wrote that any doubt that is not evenly balanced is not called a doubt, and one should follow the majority. According to what I said above, the term “evenly balanced doubt” does not describe a 50% doubt, but some vague range around 50%. Therefore, theoretically, even a fixed case in which there is clearly a majority to one side would be considered an evenly balanced doubt that one should be stringent about (even though ostensibly most chances favor leniency).

Sorry for the length.
I’d be glad to hear what you think about this.

One more point I forgot, regarding the rule: anything that separated is presumed to have separated from the majority. According to what I said above, there would also be room to treat even an 80% doubt as an evenly balanced doubt (because perhaps from the standpoint of plausibility it counts as evenly balanced). It’s just that the Sages said that one may treat a doubt involving separation from the majority as a “probabilistic” doubt, and that is a novelty. Without this novelty, there would have been reason to think that this doubt too should be treated as a “plausibility” doubt like any other doubt, and then it would count as an evenly balanced doubt. In a fixed case, that novelty does not exist, and therefore it should be treated as a “plausibility” doubt like any ordinary doubt. If we are speaking, say, about a doubt of 1 in 1000, then clearly that cannot be considered an evenly balanced doubt even in terms of plausibility; rather, the Sages were speaking about the common case of plausible doubts (which lie in some vague range around 50%), whose law is that they count as evenly balanced doubts (half-and-half).

It is clear that one cannot quantify a doubt in terms of levels of plausibility. But I don’t think this is a fundamentally different type. There is a technical problem that there is no event space whose parts can be measured and assigned numbers (percentages). But even in unquantified doubts, as they multiply, plausibility decreases. If the plausibility in my eyes that there is a God is at a certain level, and the plausibility that if He exists then He gave the Torah is at a certain level (perhaps different), one still has to multiply them by each other.

Also factually, I explained in one of the columns about following the majority in a religious court, and I showed that even there one cannot quantify the chance that the majority is correct, and still we follow the majority. In cases of a majority that is before us, one can generalize on the basis of a sample, but in a religious court even that cannot be done. It is simply a matter of plausibility.

I also do not see the connection to the question of separation and fixed cases. Why is a fixed case plausibility and not probability? When you took a piece of meat from a certain store, the chance that it is a non-kosher store is 10% (assuming there is one out of 10 in the city). I do not see why separation is different from a fixed case in this regard.