Is this a logical fallacy? There exists unique x ∈ A. Therefore, there are some x ∈ A

GENERAL

There exists exactly one x ∈ A. Therefore, there are some x ∈ A.

EXAMPLE 1

• CHILD: Sometimes you forget to pick me up from school. Therefore, I want to start walking home.
• PARENT: I have only forgotten to pick you up one time from school, and that was 3 years ago.

EXAMPLE 2

• CHILD: I want to make an omelette
• PARENT: There are some eggs in the fridge
• CHILD: There is one egg in the fridge.

In logic and mathematics, "some" means "one or more".
The fallacy might be that the word "some" in English is not the same as word "some" in logic and mathematics.
This might be viewed as a specialized variant of the fallacy of equivocation.

Imagine trying to translate "This valley is full of mist " into German.
"Dieses Tal ist voller Mist " actually translates to "this valley is full of shit."

Both German and English use the word "mist".

• In English, "mist" means water vapor / fog.
• In German, "mist" means manure.

If the English word "or" is inclusive, why does English contain the phrase "and/or"? Would not simply writing "or" be sufficient?

What if Logic and English are two different languages, analogous to how German and English are different languages? Is translating English "some" into logical "some" a mistake?

• Why ? In logic "there are some..." is symbolized with the existential quantifier, that reads "there is at least one..." This does not contradict "there is exactly one". – Mauro ALLEGRANZA May 26 '20 at 6:13
• @MauroALLEGRANZA But it is true that in plain English, "some" means more than one (that's why plural is used) while in logic, it is often translated into the existential quantifier. But then, natural language and logic are two different languages. Which is no problem, since serious logicians usually know and are aware of that. – Philip Klöcking May 26 '20 at 7:51
• Maybe useful Generalized Quantifiers – Mauro ALLEGRANZA May 26 '20 at 8:11
• You are confusing real world languages with Mathematical logic. To be exact in Mathematical logic the context SOME is used in means AT LEAST ONE. In normal conversation people may take the context to mean more than one. But then again context is everything in communication. Some things are supposed to be left in the math class. You will discover many ideas do not correspond to the real world outside of your math classes. You are taking things too seriously and not acknowledging context if you are in this scenario. Context over math in the real world. They are not always the same thing. – Logikal May 27 '20 at 14:49

You're asking if the inference from There is one A to There are some A's is a logical fallacy. The answer is no, but there is a reason why some seems to mean more than one.

First, note that in your Example 1 the parent can seem to contradict the child even if the parent agreed that there was more than one time:

Child: Sometimes you forget to pick me up from school. Therefore, I want to start walking home.

Parent: I have only forgotten to pick you up twice from school, and that was 3 years ago.

But if the parent can disagree with the child by saying this, does that mean that sometimes means more than twice? No. That's because of the pragmatic, rather than semantic, meaning of the child's assertion. What the child communicates in this context is that the parent forgets to pick the child up from school often enough to warrant walking home instead. That isn't part of the semantic meaning of sometimes, but that's what the child's usage conveys in this context.

The meaning that the speaker conveys that goes beyond the semantic meaning of the words they're using is an implicature. Implicatures are very common in everyday speech. Here is another example using the word some:

A: I ate some of the cookies.

B: (looks at the empty plate) You ate all of them!

Here A's statement conveys the implicature that A didn't eat all of the cookies, although that isn't explicitly stated. Because false implicatures are misleading, B objects to it, even though, again, it's not part of what A said.

How can we tell that the above implicature isn't part of the explicit meaning of A's statement? The answer is that implicatures can be cancelled, for instance:

C: Did A eat some of the cookies?

B: Yes, A definitely ate some of them. In fact, he ate all of them!

Here B's fist part of the statement conveys the implicature that A didn't eat all of the cookies. But the second part cancels it. If not all were part of the semantic meaning of some, B's answer would be a contradiction. But it isn't.

The same goes for some and one. The word some typically conveys the implicature more than one, but not always, and when it does, it can be cancelled:

A: Did you try some of the cookies?

B: Yes, I tried one.

Here B cancels the implicature of more than one. If some logically implied more than one, B's answer would be tantamount to "Yes, no".

One reason why implicatures are generated is that speakers are expected to say things that are relevant and reasonably informative. If you know, for example, that there's only one cookie but say that there are some cookies, you're not being as informative as you can. A hearer would take your statement to convey that there's more than one, not because of logical entailment, but because of the expectation to abide by the norms of conversation. The pragmatic inference goes something like this: if there was just one cookie, you would have said so, but you said some, so there must be more than just one.