What is the Rabbi’s view regarding how strict one should be on this issue? In your opinion, can one be more relaxed about it relative to the standard required by most halakhic decisors?

Regarding unintentional action, this probably depends on the dispute between the Taz and Rabbi Akiva Eiger about a doubtful inevitable result, since if there is a worm inside the fruit then it is an inevitable result that you will eat it.
More likely, this is a case of mere occupation with the act, in which case inevitable result is irrelevant. But even here it’s not clear to me that this really is mere occupation with the act, since at least with certain species the concern about worms is known, and people who are aware of it cannot say, “I wasn’t aware of this at all.”

The reasoning of “since he derived benefit” seems to me not relevant here at all. Both because there is no benefit from worms, and because they are tiny and negligible (the taste is nullified in sixty, and even if it is a complete creature, regarding taste that doesn’t matter. Benefit is a factual matter).

I don’t know what standard is required by most halakhic decisors. I think an initial inspection is certainly appropriate (because where it is possible to avoid the issue, it is preferable to do so. Especially if one takes into account the rabbinic claim that forbidden foods dull the soul). But I definitely would not get hysterical about this inspection. Certainly not with fruits that are not known to be prone to infestation.

There are people who try to avoid eating things that are insect-prone in kosher restaurants/hotels and when they are guests at religious people’s homes who probably did not do a meticulous inspection (we’re talking about things found in lots of foods: lettuce, cabbage, dried figs, strawberries, parsley, cauliflower, broccoli, etc.). The question is whether this is a proper practice or an excessive stringency.

It is a good practice, but in my opinion not necessary.

Does the reasoning of “its ways are ways of pleasantness” apply to insects? That is, to forbid certain fruits because of tiny insects that ordinary people are not put off from eating—wouldn’t it be contrary to “ways of pleasantness” to forbid a fruit altogether in such a case? Together with all the other reasons to permit, since biblically it is nullified in the majority and therefore the whole prohibition is only rabbinic, and there is doubt in many senses—maybe it is nullified in the mouth, maybe it isn’t there, since we did not find it, etc. etc.

There is room for that, at least according to some opinions. I’m copying a passage from Wikipedia about “ways of pleasantness”:
There are those who wrote that the prohibition on commerce in new grain may be permitted on the grounds of “her ways are ways of pleasantness,” because when the Temple stood, new grain was common, and it is not a way of pleasantness for trade in it to be impossible. Rabbi Shmuel Wosner wrote that one should not permit prohibitions on the grounds of “her ways are ways of pleasantness” when this is not explicitly stated in the Talmud. However, from the words of the Radbaz, the Chacham Tzvi, and others, it emerges that they do indeed treat the reasoning of “ways of pleasantness” as a principle that generates law.

Regarding what you wrote above:
“As for unintentional action, this probably depends on the dispute between the Taz and Rabbi Akiva Eiger about a doubtful inevitable result, since if there is a worm inside the fruit then it is an inevitable result that you will eat it.”

I thought about this issue again, and an idea occurred to me that perhaps there is no inevitable result or doubtful inevitable result here after all, because one can always attribute it to the insect being in a different place in the food with each bite, and then the certainty required for an inevitable-result act is missing. Meaning, if I’m walking in a field, and the chance of uprooting a blade of grass with each step is, say, 50%, then taking one step is not considered an inevitable result. But crossing a field in 100 steps might be considered an inevitable result, because there is a very high certainty that some grass will be uprooted. My claim is that even crossing a field in 100 steps is not an inevitable result, because each individual step is not an inevitable result, and therefore the whole aggregate is not one either. Only in an act that cannot be divided into smaller units—like cutting off a chicken’s head—does inevitable result apply. Since eating foods can be divided into small portions, and one can always say that in some small portion there is no absolute certainty that an insect is there, then even in eating the whole food there is no inevitable result. According to this, seemingly by strict law there is no obligation to inspect even a vegetable that is heavily infested with insects, like lettuce, because one is “not intending” to eat creeping creatures. What do you think?

Possibly (it seems this depends on the sugya of the two paths). But in a place where there is no way to eat just one bite of an apple, and eating the whole apple is perceived as one act, perhaps there is no room for this reasoning. You are looking for permission to eat the apple, not to take a bite from the apple.
Perhaps this is the dispute between the Tannaim regarding palm-fiber brooms (Shabbat 124b). The one who says it is not an inevitable result probably understands that there is no necessity that with every sweep of the broom a leaf will be detached. And the other perhaps argues that in the overall sweeping it is clear that one will be detached.
If I am right about this, then Rabbi Shimon Shkop’s brilliant proof collapses. He proved from here (Sha’arei Yosher, part 3, chapter 26) that doubtful inevitable result is a matter of reality and not of knowledge (like Rabbi Akiva Eiger against the Taz), because the Tanna who disagrees and says this is an inevitable result knows that his colleague thinks it is not. So if the matter depends on knowledge and not reality, then from the colleague’s perspective too it should be permitted. From here he proves that it is a matter of reality and not knowledge.
But according to our approach here, this is a halakhic dispute and not a factual one (whether the leaf must necessarily fall or not). And in fact, according to the Taz, one may have to say this, otherwise he is directly contradicted by the Mishnah.
I discussed this at length in chapter 4 of book 12 in the Talmudic Logic series (Fuzzy Logic in the Talmud).
But one still has to discuss my reasoning here: whether sweeping the whole house is considered one act, or whether that is worse than eating an apple.

I will explain what he wrote in parentheses (“it seems this depends on the sugya of the two paths”).
Regarding someone who violates a prohibition in a “whichever way you look at it” situation on both sides of the doubt, he wrote in Sha’arei Yosher 1:7 that one who walked on one path, prepared pure foods, was sprinkled, immersed, then walked on the second path and prepared pure foods—is liable (and he rejected the proof of the Pri Chadash from one who walks on the two paths one after the other and prepares pure foods, that he is liable, because impurity is a “reality” that generates the law, and reality does not disappear because of doubt; rather there is only a legal permission regarding the prohibitions, and therefore if the reality is certain in a “whichever way you look at it” way, then it certainly generates the prohibitions).
That means maybe it is permitted to eat the apple bite by bite, but he still committed a transgression (I don’t know how to formulate the practical difference, but I have a feeling there is one). And it seems this is even without Rabbi Shlomo Zalman’s view that in every doubtful prohibition that is biblically permitted, if one nonetheless did it he “requires atonement” (and then he strains greatly to explain Maimonides in places where doubtful prohibition was explicitly permitted, like orlah and a mamzer, saying that although it is permitted like any doubtful prohibition, if one did it he requires atonement, and reason dictates not to eat—but the explicit permission comes to teach that this is a scriptural decree that one may eat, and we should not outsmart it by acting according to what reason dictates. I never managed to understand his view there).
The reasoning that the apple is one act (that is, regarding the prohibition, and not only regarding the consequences, such as being “liable” for preparing pure foods) is a separate issue.

But maybe with the apple there is a majority for each bite that there is no worm in it, and then that is already a different matter.

Can my reasoning be used to be lenient and eat sushi in restaurants containing nori seaweed, which I understand is infested but hard to inspect?
Here is an illustration of the infestation and the difficulty:

It’s hard for me to say. In general, there is room to be lenient with such tiny worms even without your reasoning, so perhaps it can be added. If the seaweed is ground up there, it is easier.

Why is there room to be lenient with tiny worms? Because the Torah was not given to ministering angels?

The seaweed is not ground up, but rolled into thin sheets and then dried.

Yes. The reasoning is that what the eye does not control is not included in halakhic prohibitions. As stated, this is debatable. Especially because here, as I understand it, with effort one can see them. But when they are not seen, this is a case of mere occupation with the act (and in my opinion the rule of mere occupation with the act in forbidden fats and forbidden sexual relations does not apply here, because one does not sense the taste and derives no benefit from it), or of unintentional action, and that too has already been discussed (as noted, it depends on doubtful inevitable result).

Regarding what you wrote above, that my reasoning depends on the law of the two paths or on the question whether eating the apple is considered one act, I’ll sharpen it further. Suppose the chance that there are insects in the apple is 50% for every cubic centimeter of apple. Then in one bite the probability is indeed not certain (assuming a bite is on the order of a cubic centimeter). But even regarding the whole apple, the probability is still not certain, but say 95%. In that case I claim that this is not an inevitable result but “unintentional action.” In other words, inevitable result is defined as a situation where there is absolute certainty that one cannot do the act without a transgression resulting—like cutting off a chicken’s head on the Sabbath. But if there is even a small chance that no transgression will result, that is enough to classify the act as “unintentional.” Therefore I think that regardless of the law of the two paths, or the question whether eating the apple is considered one act or not, this is still a case of “unintentional action.”

I didn’t understand. We already got past the question of doubtful inevitable result. We are now discussing the view that doubtful inevitable result is forbidden. So even if there is no certainty, if there are worms in the apple this is a doubtful inevitable result. If you follow the Taz, you don’t need any arguments. It is permitted, period.

My claim is that there is no inevitable result or doubtful inevitable result here. This is a different situation of “unintentional action.” Doubtful inevitable result refers to a case of uncertainty in the state of affairs, not uncertainty in whether the action will drag a transgression along with it. For example: when I close a box and there is doubt whether or not there is a fly inside, there is no doubt regarding whether closing the box will cause trapping, because if there is a fly here, trapping will certainly occur. The doubt is whether there is a fly here or not. In the case of pulling grass on the Sabbath or eating an apple with worms, there is doubt as to whether the walking or the bite will cause the transgression of uprooting or of eating creeping creatures. So the doubt is not about the background state but about the action itself—whether it will entail a transgression or not. Therefore my claim is that regardless of all the disputes you mentioned, this is a case of “unintentional action,” which everyone agrees is permitted, as Maimonides writes:
“Things that are permitted to do on the Sabbath, and while doing them it is possible that a labor may be done because of them and possible that no labor may be done—if he did not intend that labor, it is permitted.”

[Just to note: a nazirite is forbidden to comb his hair because of the inevitable result of pulling out hair, even though there is no doubt in the background situation, and not every single act of combing will certainly pull some out.]

On the contrary, I looked inside at Nazir 42a and saw there that it is discussed as “unintentional action” and not as doubtful inevitable result. Correct me if I’m wrong.

Oren,
If it is not certain that the apple contains insects, then it really is unintentional action. So what is the question? Why do you need the bite-by-bite argument?

What I meant is that even though some vegetable (an apple for the sake of argument) has a presumption of infestation, that presumption is not certain (say 99%). Only where there is absolute certainty, as in cutting off a chicken’s head so that it will die, is the act defined as an inevitable result. But if there is even a slight possibility that no transgression will occur, then this is “unintentional action.” And that also fits the wording of Maimonides that I cited above:

“Things that are permitted to do on the Sabbath, and while doing them it is possible that a labor may be done because of them and possible that no labor may be done—if he did not intend that labor, it is permitted.”

I lost you.
If it is not certain that there is a worm in the apple, then according to Rabbi Akiva Eiger this is exactly the case of doubtful inevitable result (since if there is a worm in it, I will certainly eat it—aside from your bite argument). This is exactly the dispute over whether there is permission of unintentional action in eating such an apple. And on this you raised the bite argument, no? Doubt whether there is or is not a worm in the apple parallels doubtful inevitable result, not doubt whether a prohibited act will result (leaving aside your bite argument).
Maybe I missed something, but I don’t understand.

You’re right, I got mixed up.

But still, something is unclear to me regarding the Talmud in Pesachim 25b, which says that benefit that comes to a person against his will is considered unintentional action and not inevitable result. But there there is no doubt that the person will benefit from the smell of the idolatry in the marketplace, so why is that not defined as an inevitable result?

I’m not currently immersed in the sugya. The Gemara itself deals with the connection between this and the dispute between Rabbi Yehuda and Rabbi Shimon. It seems to me that the focus is that he has no other option (it is unavoidable).

[Oren. I wrote from memory and I’m not immersed in the matter.
At first glance it seems that Rabbah in Nazir 42a does not hold that Rabbi Shimon admits inevitable result (rather only in the next generation, Abaye and Rava. Therefore Rabbah explained that anyone combing intends to remove loose hairs).
But see Shabbat 81b: the halakha follows an anonymous Mishnah, and we learned, “A nazirite may wash and separate his hair with his fingers, but may not comb,” and they explain: “but may not comb with a comb, because it definitely pulls out hair,” and Rabbi Shimon admits “inevitable result—and shall he not die?” But washing and separating with the fingers—sometimes one does not pull hair out, and when he does, it is a permitted unintentional act.]

[And one last intrusion. Regarding smell—and again from memory, because I’m not immersed and if I’m mistaken, fair enough—in the case of craving aroma they disputed whether smell is considered substantial. There are nice discussions of this in Rabbi Shlomo Zalman’s novellae to Nedarim and in the Ran there, who apparently says that there is no inevitable result in deriving benefit from smell, only in food and sexual prohibitions; see there.
https://hebrewbooks.org/pdfpager.aspx?req=40915&st=&pgnum=20]

Correct link:
https://hebrewbooks.org/pdfpager.aspx?req=40915&st=&pgnum=20

I googled a bit about the issue of inevitable result in prohibitions other than Sabbath prohibitions, such as prohibitions of benefit, and found the following:

Tosafot and the Ran, in the name of the Ra’ah, write in tractate Shabbat that if a person will inevitably derive benefit from the prohibition, one must forbid him from deriving that benefit, and he may not walk in a place where he will inevitably benefit from the prohibition. The Rosh agrees as well, and explains that when walking past a house of idolatry where fragrant incense is offered to idolatry, this is not an “inevitable result” that he will benefit from the prohibition, because he can stop up his nose. Similarly, Rabbi Abraham son of Maimonides writes in his book Birkat Avraham (section 9) that this is not an “inevitable result” that he will benefit from the smell, because perhaps a wind will come and disperse the smell, or perhaps it will mix with a bad odor that will spoil its pleasant smell.

The Maharam Chalava takes a different view. He holds that when a person does not intend to benefit from the smell, this is not defined as an “inevitable result,” even though the smell enters his nose, because the pleasure of smell is substantial only when a person intends to enjoy it; but when he enjoys it without intention, it is not full enjoyment, and therefore is not forbidden.

A different approach is that of the Riaz (in his rulings), who holds that with prohibitions of benefit one does not say the rule of “inevitable result” at all; rather it depends only on intention. If he intends to benefit, it is forbidden; and if he does not intend to benefit, it is permitted even though this is an “inevitable result.” Similarly, the Ran in tractate Chullin (32a in the Rif pagination) writes that regarding benefit, “even though it is an inevitable result—it is permitted.” However, he qualifies this by saying that in benefit through eating and forbidden intercourse, one should forbid “inevitable result,” based on the rule that “one merely occupied with forbidden fats and forbidden sexual relations is liable, since he derived benefit.”

For the view that permits “inevitable result” in prohibitions of benefit, there is proof from a Mishnah in tractate Kilayim (9:5), where we learned that a seller of garments made of forbidden mixtures may wear them for the sake of selling them if he does not intend to benefit from them, even though there seemingly it is an “inevitable result” that he will benefit from the garment; and the Gemara in tractate Shabbat (29b) establishes this Mishnah according to Rabbi Shimon, who permits a matter that is unintentional. However, Tosafot in tractate Shabbat explain (s.v. “provided that”) that this is a case where it is not an “inevitable result” regarding the benefit, such as where he wears other garments underneath them, or wears them on his shoulders.
———-

Seemingly, based on this there is room to doubt whether inevitable result applies at all to insect prohibitions. In other words, there is a double doubt here. The first doubt is whether inevitable result applies to insect prohibitions at all, and the second doubt is whether the halakha follows the Taz that doubtful inevitable result is permitted like “unintentional action.” According to this, seemingly the matter is permitted from the outset, no?

This is a case of eating, and even according to the Riaz there is inevitable result in prohibitions of eating.

For him there is inevitable result in prohibitions of eating only where one derives benefit (“one merely occupied with forbidden fats and forbidden sexual relations is liable, since he derived benefit”), but above you wrote that with insects the reasoning of “since he derived benefit” does not apply, because there is no benefit here or it is completely imperceptible.

According to that, the worm is nullified because it gives no taste. A complete creature is not nullified even though it gives no taste, and therefore there is still a prohibition of eating here. I don’t think he distinguishes between a prohibition of eating that involves benefit and a prohibition of eating that does not. If the worm is not nullified, then the benefit is from the apple + the worm, and that is still benefit from a prohibition.

How does that fit with what you wrote above:
“The reasoning of ‘since he derived benefit’ seems to me not relevant here at all. Both because there is no benefit from worms, and because they are tiny and negligible (the taste is nullified in sixty, and even if it is a complete creature, regarding taste that doesn’t matter. Benefit is a factual matter).”

Maimonides rules in the laws of forbidden foods 14:12:
“One who eats a food from among the forbidden foods jokingly or while merely occupied with the act, even though he did not intend the act of eating itself—since he derived benefit, he is liable just as one who intended the very act of eating; and benefit that comes to a person against his will, from any of the prohibitions—if he intended it, it is forbidden, and if he did not intend it, it is permitted.”

From Maimonides’ words “since he derived benefit,” it sounds like benefit is a necessary condition in order to be liable for eating while merely occupied with the act.

As I recall, I wrote above that there is definitely room to permit. But it is hard for me to reach a conclusion that way. This needs to be thought through.
In this discussion I am raising possibilities, not making definitive statements or issuing a halakhic ruling. I raised here the possibility that even if you derive no benefit from the worm itself, if it is not nullified in the apple, still, since there is a prohibition of eating, this is also considered benefit, and perhaps the reasoning of “since he derived benefit” applies after all (from the apple + worm). Especially according to Maimonides, for whom prohibitions of eating are just another type of prohibition of benefit (eating is only an example of a common form of benefit), it is hard to accept that there are prohibitions of eating that would not count as prohibitions of benefit, and therefore there is reason to say that “since he derived benefit” applies to them.
Two halakhot before the halakha you cited, Maimonides speaks about the fact that one is liable for eating only when one eats in the normal manner of deriving benefit. So why would one be liable for eating a worm that is not nullified? (I mean when he does intend it.)