As someone who is now mostly working in constructive / intuitionistic logic ($\mathsf{IL}$) I am still wondering about the most concise way to spotlight the relevance to people who so far only know about classical logic.
Most importantly I think it helps to give a convincing example showcasing why double negations cannot in general be removed. Before being aware of $\mathsf{IL}$ I definitely removed all seemingly unnecessary negations in any sentence I heard, just to ease comprehension. For example I would have read the sentence
It's not wrong to say that I regularly removed too many negations.
as
I regularly removed too many negations.
without suspecting that this could change the meaning. (After reading a bit it does also seem to depend on the language you speak, whether double negations cancel or not.)
My first questions are therefore:
- Do you have a go-to example to illustrate why adapting $\mathsf{IL}$ is useful, even in "every day" logical endeavors?
- Are there any "natural language" sentences using a double negation which would have a different meaning when we remove just them?
A related second issue is about the reading of double negations. It is accurate but still a bit cumbersome to read $\neg \neg$ as "it is not wrong that ...". Taking the following example as a benchmark: $\neg \neg A \to \neg \neg (A \to B) \to \neg \neg B$ we get
If it is not wrong to assume $A$ and it is not wrong that $A$ implies $B$, then it is not wrong to conclude $B$.
I have seen one author use the word almost to refer to $\neg \neg$ and I think this eases readability a bit:
If we almost have $A$ and we almost have that $A$ implies $B$, then we almost have $B$.
But I imagine their being even better fits, which prompts my third question:
- What is a pleasant and accurate word to read $\neg\neg$ by?